Revista Electronica MateInfo.RO ISSN 2065-6432
MAI 2010
(format HTML)
www.MateInfo.ro
(1175 -1240)
Din nou despre sirul lui
Fibonacci
NECULAI STANCIU
Abstract
The
purpose of the article is to describe the contributions to Mathematics made by
the thirteenh century Italian, Fibonacci.Unfortunately, not much is known about
Fibonaccci’s personal life.Representative problems solved by Fibonacci are set
as challenges to the reader.
After
a brief historical account of Leonardo Pisano Fibonacci, some basic results
concerning the Fibonacci numbers are developed and proved, and entertaining
examples are described.Connections are made between the Fibonacci numbers and
the Golden Ratio, biological nature, and other combinatorics examples.
We
are considering both the originality and power of his methods, and the
importance of his results, we are abundantly justified in ranking Leonardo of
Pisa as the greatest genius in the field of number theory who appeared between the
time of Diophantus and Fermat.
Key words: History of Mathematics, Fibonacci’s Rabbits, Fibonacci numbers and
nature,Divine proportion, Golden Section in Art(Architecture, music and human
body), The Fibonacci sequence, Fibonacci identities, matrix methods.
M.S.C.: 01-XX, 01AXX, 01A05, 11B39,
11B37, 11B50.
1. Istorie.
1.1. Cine a fost Fibonacci?
Fibonacci(1175-1240) a fost unul dintre cei mai mari matematicieni ai evului
mediu.S-a nascut in Italia, in orasul Pisa, faimos pentru turnul sau inclinat,
care parca sta sa cada.
Tatal
sau, Bonacci Pisano, a fost ofiter
vamal in orasul Bougie din Africa de Nord , astfel ca Fibonacci a crescut in mijlocul civilizatiei nord-africane.A
cunoscut astfel multi negustori arabi si indieni (deoarece a facut multe calatorii
pe coastele Mediteranei) de unde a deprins stinta lor aritmetica, precum si
scrierea cifrelor arabe.
1.2. Cartile lui Fibonacci.
In 1202 revine in Italia unde publica un
tratat de aritmetica si algebra intitulat “Incipit
Liber Abacci”( compositus a Leonardo
filius Bonacci Pisano).In acest tratat introduce pentru prima data in
Europa sistemul de numeratie arab, cifre pe care le folosim si in zilele
noastre:0,1, 2, 3,…,9.
In
1220 publica “Practica Geomitriae”,
un compendiu de rezultate din geometrie si trigonometrie, apoi in 1225 “Liber Quadratorum” in care studia
calculul radicalilor cubici.Cartile lui Fibonacci
au cunoscut o larga raspandire asa incat timp de peste doua secole au fost
considerate sursele cele mai competente in domeniul numerelor.
Pentru
a intelege mai bine situatia din acele vremuri trebuie sa aruncam o privire pe
matematica in Europa si in Orient.
Matematica araba
Imperiul
arab, odata cu aparitia Islamului (sec VII), se extinde foarte repede cuprinzand
Orientul Apropiat, o parte din Asia Mica si Centrala, ajungand pana la Valea
Indului, nordul Africii si Peninsula Iberica.Se ridica importante centre
culturale ca:Bagdad, Samarkand, Buhara, Horezm, Damasc, Cordoba, Granada,
Sevilla, Toledo, dupa ce in prealabil fusese distruse Ispahanul, Persepolis si
Alexandria.
Matematica
araba este matematica creata sub dominatia araba, nu neaparat apartinand
arabilor, deoarece putini dintre matematicienii arabi erau de origine araba
dar, au asimilat foarte repede cultura Orientului precum si cea Elena pe care
le transmit in diverse parti ale imperiului.Primul mare matematician arab a
fost Al-Horezmi (780
–
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@
850).Din opera sa se detaseaza “Algebra” structurata pe 4 capitole
(Solutiile ecuatiilor,Calculul dobanzilor, Geometria, Algebra testamentara).Al-Horezmi a fost primul matematician
care a stabilit reguli pentru adunare, scadere, multiplicare si divizare cu
noile numere arabe.De la el provine cuvantul algoritm (incercati sa spuneti numele Al-Horezmi repede de cateva ori!).Intr-un alt tratat “Stiinta transpunerii si a reducerii”, specifica procesul manipularii
ecuatiilor algebrice, “al-jabr”, a
ajuns la noi ca algebra.
Abu Kamil (900), nascut in Egipt, este continuator a lui Al-Horezmi.In “Cartea raritatilor
din aritmetica” se ocupa cu rezolvarea in numere intregi a sistemelor
liniare nedeterminate.
Abu Wafa (940
–
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@
997) s-a ocupat cu geometria practica.In
lucrarea sa “Cartea perfecta” expune
bazele trigonometriei, inclusive teorema sinusurilor.De asemenea rezolva
probleme de trigonometrie sferica, utilizand cu predilectie functia cotangenta.
Al-Hazem (1000) prin “Cartea opticii”
este un precursor al acestei stiinte.Tot el formuleaza axioma lui Pasch si incearca demonstrarea
postulatului V al lui Euclid.
Omar Al-Khayyam (1048
–
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@
1123), este primul matematician care expune o
teorie generala a ecuatiilor de gradul III.Recent a fost descoperit un memoriu al sau asupra
operei lui Euclid. Omar Al-Khayyam, conducatorul
Observatorului astronomic din Ispahan, s-a ocupat si cu “patrulaterul Saccheri” (care, de drept ar trebui numit patrulaterul lui Omar), apoi a dat prima
formulare a axiomei lui Arhimede.Era
vestit si ca poet.
Al-Biruni (973
–
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@
1048), persan de origine, este cel care in
1030 introduce cercul trigonometric.Tot el calculeaza lungimea meridianului
terestru la 41.550 km.
Nassir ed Din al Tusi (1201
–
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@
1274), conducatorul Observatorului astronomic
din Maraga, s-a acupat cu teoria paralelelor.In “Tratatul despre patrulaterul complet” a facut o expunere integrala
a rezolvarii triunghiurilor (plane si sferice).
Al-Kashi (1400), iranian de origine, in “Cheia
aritmeticii” se ocupa cu formula binomului si cu extragerea de radacini.S-a
ocupat intens si de calcule aproximative, iar in “Tratatul despre circumferinta” din 1424, da valoarea numarului
π
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWdahaaa@37B0@
cu 16 zecimale exacte.
De
aici, matematica si in general cultura araba decade.
Matematica evului mediu.
Cruciadele
(campanii pentru recucerirea locurilor sfinte), prilejuiesc stabilirea de legaturi
cu cultura araba musulmana (mai ales in Spania si Sicilia) precum si cu Bizantul.
Dupa ce Spania este recucerita de
mauri, Toledo devine centru cultural de prestigiu.In acest moment incep
traducerile din araba.Printre primii traducatori este englezul Adelard de Bath (1100) care, deghizat ca
student mahomedan la Cardoba traduce din limba araba “Elementele” lui Euclid si
“Algebra” lui Al-Horezmi, iar din limba greaca, opera lui Ptolemeu.Din aceeasi perioada se remarca si alti traducatori ca: Ioannes din Sevilla si Gerardo din Cremona (1114
–
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@
1187) care au tradus circa 80 de lucrari
clasice din limba araba .Ambii au lucrat la Toledo.
Secolele XII
–
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XV reprezinta perioada de asimilare a matematicii antice si a celei orientale.
Leonardo
da Pisa, este pe drept considerat primul
mare matematician original al Europei.In numeroasele sale calatorii (Egipt,
Siria, Grecia, Sicilia) ia contact cu cu cultura elena si cea araba .
LIBER ABACI
–
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbeqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C1@
O CARTE REMARCABILA
Povestea
numerelor apare in Italia in 1202, o data cu aparitia cartii Liber Abaci, scrisa de Leonardo Pisano, pe atunci in varsta de
27 de ani.Cartea, are 15 capitole, si sunt scrise in intregime de mana, tiparul
aparand 300 de ani mai tarziu.Leonardo,
a fost inspirit sa scrie cartea dupa o vizita la Burgia, un oras prosper
Algerian, unde tatal sau era consul de Pisa.In acest timp,Fibonacci a invatat secretele sistemului de numere indo-arab, pe
care arabii l-au introdus in Vest in timpul cruciadelor.
Cartea
a atras numerosi adepti in randul matematicienilor din Italia, precum si din
restul Europei.Liber Abaci, a dezvaluit
oamenilor o cu totul alta lume, unde numerele au inlocuit literele.Fibonacci incepe cartea cu notiuni
despre identificarea numerelor, de la unitati la cifra zecilor, a sutelor, a
miilor etc.In ultimile capitole gasim calcule cu numere intregi si fractii,
regulile proportiilor, extrageri de radacini patrate si de ordin superior, apoi
se prezinta solutiile ecuatiilor liniare si patratice
Liber Abaci era plina cu exemple practice:calcule de contabilitate financiara,
calculul profitului, schimbul de bani, conversia greutatilor, calculul imprumutului
cu dobanda (interzis in acel timp in diverse locuri ale lumii).
Desi era cunoscut in anul 1000, si desi Liber Abaci a explicat avantajele ,
sistemul de numarare, indo-arab, nu a prins la scara mare pana aproape in 1500
e.n.Motivele au fost, in mare parte doua.Primul tine de inertia umana si
rezistenta la schimbare a omului, pentru ca invatarea unui sistem radical nou
cere timp si de faptul ca biserica catolica din acea perioada considera cifrele
arabe de origine pagana.Al doilea motiv este de natura practica, deoarece era
mult mai usor sa se comita fraude.Era tentanta schimbarea lui 0 in 6 sau 9, iar
1 putea fi usor inlocuit cu 4, 6, 7, sau 9 (de atunci europenii scriu 7 cu codita!).
Desi noile numere au aparut in Italia, Florenta
a emis un edict in 1229 prin care interzicea bancherilor folosirea simbolurilor
“infidele”.Ca rezultat, multi dintre cei care voiau sa invete noul sistem se deghizau
in musulmani.
Originea sistemului de
numere.
Putem
aprecia succesul lui Fibonacci cu Liber Abaci doar daca privim cum a
evoluat societatea, din punctul de vedere al numerelor, pana la el.Masurarea si
numararea au aparut cu cateva zeci de
mii de ani inaintea lui Hristos.Oamenii au infiintat primele asezari pe
malurile Tigrului si Eufratului, Nilului, Gangelui, Indului si
Amazonului.Fluviile erau folosite pentru comert si transport, iar aventurierii
au descoperit marile si oceanele unde se varsau apele.Calatoriile pe distante
lungi cereau masurarea timpului si
calcule precise.Preotii erau de obicei astronomi, iar din astronomie a venit
matematica .
In
450 i.e.n., grecii au inventat un sistem numeric alfabetic, care folosea cele
24 de litere ale alfabetului grecesc si alte trei litere , care mai tarziu au
disparut.Fiecare numar de la 1 la 9 avea propria litera, la fel si multiplii de
10.Alfa, insemna 1, iar “ro” reprezenta 100.Astfel, 112 se scria
“ro-deca-beta”.Acest sistem se putea folosi cu greutate pentru calcule.Abacul, era cel mai vechi aparat de numarat
din istorie.
Un
occidental, matematician din Alexandria, Diofantus,
prin 250 e.n., a sugerat un sistem de numere comparative cu sistemul de
litere.Remarcabilele sale inventii au fost ignorate vreme de 1500 de ani.Pana
la urma, lucrarea sa a fost recunoscuta cum se cuvine si a jucat un rol
important in algebra secolului al XVII-lea.Ecuatiile algebrice , de forma
ax+by=c
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaadIhacqGHRaWkcaWGIbGaamyEaiabg2da9iaadogaaaa@3C8B@
,
se numesc “ecuatii diofantice”.
Piesa
centrala a sistemului indo-arab a fost inventarea lui “zero”, “sunya” la indusi,
“cifr” in araba, “tsfira” in ruseste
–
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@
ceea ce inseamna “numar”.Terminul provine de
la “cipher”, ceea ce inseamna “gol” si se refera la coloana goala de
la abac.
1.3. Sirul lui Fibonacci.Numele
Fibonacci.
Fibonacci a ramas in memoria noastra prin sirul : 0, 1, 1, 2, 3, … introdus in
anul 1202, atunci matematicianul fiind sub numele de Leonardo Pisano (Leonard din Pisa).
Mai tarziu,
matematicianul insusi si-a spus Leonardus Filius Bonacii Pisanus(Leonard fiul lui Bonaccio Pisanul).In secolul al
XIV-lea sirul prezentat mai sus a fost
denumit sirul lui Fibonacci prin
contractia cuvintelor filius Bonacii.Acest
sir apare pentru prima data in cartea mentionata mai sus “Liber Abaci”(“Cartea despre abac”), fiind utilizat in rezolvarea
unei probleme de matematica.
1.4.Iscusinta lui Fibonacci.Problema
iepurilor.Originea sirului Fibonacci.
Potrivit
obiceiului din acea epoca, Fibonacci a
participat la concursuri matematice(adevarate dispute publice) pentru cea mai
buna si mai rapida solutie a unor probleme grele(ceva in genul Olimpiadelor Nationale).Iscusinta
de care dadea dovada in rezolvarea problemelor cu numere uimise pe toata lumea,
astfel ca reputatia lui Leonardo a
ajuns pana la imparatul Germaniei, Frederik
al II-lea.La un concurs prezidat de acest imparat una din probleme date
spre rezolvare a fost: “sa se gaseasca un patrat perfect, care sa ramana patrat
perfect daca este marit sau micsorat cu 5”.Dupa un timp scurt de gandire Fibonacci a gasit numarul
1681
144
=
(
41
12
)
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaGaaGOnaiaaiIdacaaIXaaabaGaaGymaiaaisdacaaI0aaaaiabg2da9maabmaabaWaaSaaaeaacaaI0aGaaGymaaqaaiaaigdacaaIYaaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaa@41AA@
.Intradevar:
1681
144
−5=
961
144
=
(
31
12
)
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaGaaGOnaiaaiIdacaaIXaaabaGaaGymaiaaisdacaaI0aaaaiabgkHiTiaaiwdacqGH9aqpdaWcaaqaaiaaiMdacaaI2aGaaGymaaqaaiaaigdacaaI0aGaaGinaaaacqGH9aqpdaqadaqaamaalaaabaGaaG4maiaaigdaaeaacaaIXaGaaGOmaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaaaa@48E0@
si
1681
144
+5=
2401
144
=
(
49
12
)
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaGaaGOnaiaaiIdacaaIXaaabaGaaGymaiaaisdacaaI0aaaaiabgUcaRiaaiwdacqGH9aqpdaWcaaqaaiaaikdacaaI0aGaaGimaiaaigdaaeaacaaIXaGaaGinaiaaisdaaaGaeyypa0ZaaeWaaeaadaWcaaqaaiaaisdacaaI5aaabaGaaGymaiaaikdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaa@498F@
.Nu
se stie rationamentul lui Fibonacci dar,
toate incercarile, chiar si cele mai ingenioase, de a rezolva aceasta problema
cu ajutorul algebrei, duc in cel mai bun caz la o ecuatie cu 2 necunoscute.
La
un alt concurs prezidat de imparat problema propusa concurentilor suna astfel:
“Plecand de la o
singura pereche de iepuri si stiind ca fiecare pereche de iepuri produce in
fiecare luna o noua pereche de iepuri, care devine productiva la varsta de o
luna, calculati cate perechi de iepuri vor fi dupa n luni (se considera ca
iepurii nu mor in decursul respectivei perioade de n luni)”.
Solutie.Din datele problemei
rezulta ca numarul perechilor de iepuri din fiecare luna este un termen al sirului
lui Fibonacci.Intr-adevar, sa
presupunem ca la 1 ianuarie exista o singura pereche fertila de iepuri.Notam cu
1 perechea respectiva .Ea corespunde numarului
f
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIYaaabeaaaaa@37C6@
din sirul lui Fibonacci:
f
2
=
f
0
+
f
1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIYaaabeaakiabg2da9iaadAgadaWgaaWcbaGaaGimaaqabaGccqGHRaWkcaWGMbWaaSbaaSqaaiaaigdaaeqaaaaa@3D65@
=
0+1=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgUcaRiaaigdacqGH9aqpcaaIXaaaaa@3A0B@
.
La 1 februarie, mai
exista o pereche pe care o notam 1.1.Deci in acest moment sunt doua perechi ,
ceea ce corespunde termenului:
f
3
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIZaaabeaaaaa@37C7@
=
f
1
+
f
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadAgadaWgaaWcbaGaaGOmaaqabaaaaa@3A84@
=1+1=2.
La 1 martie sunt 3
perechi, doua care existau in februarie si una noua care provine de la perechea
numarul 1(se tine seama ca o pereche devine fertila dupa doua luni).Notam cu
1.2 aceasta noua pereche.Numarul perechilor din aceasta luna corespunde
termenului:
f
4
=
f
2
+
f
3
=1+2=3
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaI0aaabeaakiabg2da9iaadAgadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcaWGMbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaaGymaiabgUcaRiaaikdacqGH9aqpcaaIZaaaaa@4297@
.
La 1 aprilie exista
5 perechi si anume:trei perechi existente in luna martie , o pereche noua care
provine de la perechea 1 si o pereche
noua care provine de la perechea 1.1 care la 1 martie a devenit fertila
(pereche pe care o notam cu 1.1.1).Numarul perechilor din aceasta luna corespunde
termenului:
f
5
=
f
3
+
f
4
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaI1aaabeaakiabg2da9iaadAgadaWgaaWcbaGaaG4maaqabaGccqGHRaWkcaWGMbWaaSbaaSqaaiaaisdaaeqaaaaa@3D6E@
=2+3=5.
Termenii din aceasta
relatie se interpreteaza astfel:
f
4
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaI0aaabeaaaaa@37C8@
=numarul perechilor existente in luna
precedenta ;
f
3
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIZaaabeaaaaa@37C7@
=numarul perechilor noi(provin de la perechile
existente in luna anteprecedenta).
Procedand in
continuare in acest fel, vom deduce ca la data de 1 decembrie numarul
perechilor este dat termenul:
f
13
=
f
11
+
f
12
=89+144=233
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIXaGaaG4maaqabaGccqGH9aqpcaWGMbWaaSbaaSqaaiaaigdacaaIXaaabeaakiabgUcaRiaadAgadaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0JaaGioaiaaiMdacqGHRaWkcaaIXaGaaGinaiaaisdacqGH9aqpcaaIYaGaaG4maiaaiodaaaa@4883@
,
iar la 1 ianuarie anul urmator exista:
f
14
=
f
12
+
f
13
=144+233=377
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIXaGaaGinaaqabaGccqGH9aqpcaWGMbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabgUcaRiaadAgadaWgaaWcbaGaaGymaiaaiodaaeqaaOGaeyypa0JaaGymaiaaisdacaaI0aGaey4kaSIaaGOmaiaaiodacaaIZaGaeyypa0JaaG4maiaaiEdacaaI3aaaaa@4940@
perechi de iepuri.
Concluzia este urmatoarea
:
Daca notam cu
f
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaaaaa@37FD@
numarul de perechi de iepuri dupa
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E6@
luni , numarul de perechi de iepuri dupa
n+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgUcaRiaaigdaaaa@3883@
luni, notat cu
f
n+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaaaaa@399A@
, va fi
f
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaaaaa@37FD@
(iepurii nu mor niciodata !), la care se adauga
iepurii nou-nascuti.Dar iepurasii se nasc doar din perechi de iepuri care au
cel putin o luna, deci vor fi
f
n−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaaaaa@39A5@
perechi de iepuri nou-nascuti.
Obtinem astfel o
relatie de recurenta:
f
0
=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIWaaabeaakiabg2da9iaaicdaaaa@398E@
,
f
1
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIXaaabeaakiabg2da9iaaigdaaaa@3990@
,
f
n+1
=
f
n
+
f
n−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGH9aqpcaWGMbWaaSbaaSqaaiaad6gaaeqaaOGaey4kaSIaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaaaaa@4152@
,
care genereaza termenii sirului lui 
Fibonacci.
Observatie. Acest sir exprima intr-un
mod naiv cresterea populatiei de iepuri.Se presupune ca iepurii au cate doi pui
o data la fiecare luna dupa ce implinesc
varsta de doua luni.De asemenea, puii nu mor niciodata si sunt unul de sex
masculin si unul de sex feminin.
2. Fibonacci, numarul
de aur, natura si arta.
2.1. Fibonacci si
“numarul de aur”
Raportul
de aur este un numar irational(1.618033…), putand fi definit in diferite moduri
dar, cel mai important concept mathematic asociat cu regula de aur fiind sirul
lui Fibonacci.Impartind orice numar
la predecesorul sau, se obtine aproximativ numarul de aur.Primii care l-au
folosit au fost egiptenii, majoritatea piramidelor fiind construite tinand cont
de numarul de aur.Grecii au fost insa cei care l-au denumit astfel, folosindu-l
atat in arhitectura cat si pictura si
sculptura.Dealtfel numarul de aur se noteaza cu litera greceasca “fi”(
ϕ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dygaaa@37BB@
), de la sculptorul grec Phidias.El a construit Parthenonul pornind de la acest raport.
Sa
incepem cu o problema estetica.Sa consideram un segment de dreapta.Care este
cea mai “placuta” impartire a unui segment in doua parti?Grecii antici au gasit
un raspuns pe care ei il considerau corect(teoreticienii il numesc “simetrie
dinamica”).Daca partii stangi a segmentului ii atribuim lungimea
u=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiabg2da9iaaigdaaaa@38AE@
,
atunci partea dreapta va avea o lungime
v=0.618...
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabg2da9iaaicdacaGGUaGaaGOnaiaaigdacaaI4aGaaiOlaiaac6cacaGGUaaaaa@3DB3@
Despre un segment partitionat astfel spunem ca
este impartit in sectiunea , sau proportia sau diviziunea de aur(divina).Ideea
este ca lungimea
u
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaaaa@36ED@
reprezinta aceeasi parte din tot segmentul
(u+v)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadwhacqGHRaWkcaWG2bGaaiykaaaa@3A23@
cat reprezinta lungimea
v
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaaaa@36EE@
din partea
u
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaaaa@36ED@
.Cu
alte cuvinte:
u+v
u
=
u
v
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG1bGaey4kaSIaamODaaqaaiaadwhaaaGaeyypa0ZaaSaaaeaacaWG1baabaGaamODaaaaaaa@3CDF@
.
Daca notam
ϕ=
u
v
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dyMaeyypa0ZaaSaaaeaacaWG1baabaGaamODaaaaaaa@3AC6@
,
observam ca:
1+
1
ϕ
=1+
u
v
=
u+v
u
=
u
v
=ϕ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgUcaRmaalaaabaGaaGymaaqaaiabew9aMbaacqGH9aqpcaaIXaGaey4kaSYaaSaaaeaacaWG1baabaGaamODaaaacqGH9aqpdaWcaaqaaiaadwhacqGHRaWkcaWG2baabaGaamyDaaaacqGH9aqpdaWcaaqaaiaadwhaaeaacaWG2baaaiabg2da9iabew9aMbaa@498B@
,
si este radacina pozitiva a ecuatiei
ϕ
2
−ϕ−1=0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaWbaaSqabeaacaaIYaaaaOGaeyOeI0Iaeqy1dyMaeyOeI0IaaGymaiabg2da9iaaicdaaaa@3ECB@
,
adica
ϕ=
1+
5
2
=1.6180339887...
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dyMaeyypa0ZaaSaaaeaacaaIXaGaey4kaSYaaOaaaeaacaaI1aaaleqaaaGcbaGaaGOmaaaacqGH9aqpcaaIXaGaaiOlaiaaiAdacaaIXaGaaGioaiaaicdacaaIZaGaaG4maiaaiMdacaaI4aGaaGioaiaaiEdacaGGUaGaaiOlaiaac6caaaa@4810@
Daca presupunem
u=1,
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiabg2da9iaaigdacaGGSaaaaa@395E@
atunci :
v=
u
ϕ
=
1
ϕ
=ϕ−1=
−1+
5
2
=0.6180339887...
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabg2da9maalaaabaGaamyDaaqaaiabew9aMbaacqGH9aqpdaWcaaqaaiaaigdaaeaacqaHvpGzaaGaeyypa0Jaeqy1dyMaeyOeI0IaaGymaiabg2da9maalaaabaGaeyOeI0IaaGymaiabgUcaRmaakaaabaGaaGynaaWcbeaaaOqaaiaaikdaaaGaeyypa0JaaGimaiaac6cacaaI2aGaaGymaiaaiIdacaaIWaGaaG4maiaaiodacaaI5aGaaGioaiaaiIdacaaI3aGaaiOlaiaac6cacaGGUaaaaa@5416@
Afirmam acum ca
ϕ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dygaaa@37BB@
este strans legat de sirul lui
Fibonacci.Aceasta este o idée remarcabila a matematicii .
Mai observam ca :
ϕ=1+
1
ϕ
=1+
1
1+
1
ϕ
=1+
1
1+
1
1+
1
ϕ
=...=1+
1
1+
1
1+
1
1+
1
1+...
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@600A@
este o fractie infinita.
Daca privim fractiile
partiale :
1=
1
1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabg2da9maalaaabaGaaGymaaqaaiaaigdaaaaaaa@393A@
,
1+
1
1
=
2
1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgUcaRmaalaaabaGaaGymaaqaaiaaigdaaaGaeyypa0ZaaSaaaeaacaaIYaaabaGaaGymaaaaaaa@3BA3@
,
1+
1
1+
1
1
=
3
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgUcaRmaalaaabaGaaGymaaqaaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIXaaaaaaacqGH9aqpdaWcaaqaaiaaiodaaeaacaaIYaaaaaaa@3E0D@
,
1+
1
1+
1
1+
1
1
=
5
3
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgUcaRmaalaaabaGaaGymaaqaaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaaGymaaaaaaaaaiabg2da9maalaaabaGaaGynaaqaaiaaiodaaaaaaa@4078@
,
1+
1
1+
1
1+
1
1+
1
1
=
8
5
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgUcaRmaalaaabaGaaGymaaqaaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaaGymaiabgUcaRmaalaaabaGaaGymaaqaaiaaigdaaaaaaaaaaaGaeyypa0ZaaSaaaeaacaaI4aaabaGaaGynaaaaaaa@42E5@
,
1+
1
1+
1
1+
1
1+
1
1+
1
1
=
13
8
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgUcaRmaalaaabaGaaGymaaqaaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIXaGaey4kaSYaaSaaaeaacaaIXaaabaGaaGymaiabgUcaRmaalaaabaGaaGymaaqaaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIXaaaaaaaaaaaaaaacqGH9aqpdaWcaaqaaiaaigdacaaIZaaabaGaaGioaaaaaaa@4606@
obsevam ca toate rezultatele sunt rapoarte de
numere Fibonacci, fapt ce motiveaza teorema care spune ca :
lim
n→∞
f
n+1
f
n
=ϕ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHEisPaeqaaOWaaSaaaeaacaWGMbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaaaOqaaiaadAgadaWgaaWcbaGaamOBaaqabaaaaOGaeyypa0Jaeqy1dygaaa@45FA@
.In
cuvinte putem spune ca, pe masura ce
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E6@
se apropie de infinit, raportul termenilor al
n+1−lea
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgUcaRiaaigdacqGHsislcaWGSbGaamyzaiaadggaaaa@3C31@
si al
n−lea
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgkHiTiaadYgacaWGLbGaamyyaaaa@3A94@
din sirul lui Fibonacci se apropie de
ϕ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dygaaa@37BB@
.
La
fel de simplu cum
ϕ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dygaaa@37BB@
este o fractie infinita, tot asa poate fi si
un radical infinit:
ϕ=
1+
1+
1+
1+
1+...
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dyMaeyypa0ZaaOaaaeaacaaIXaGaey4kaSYaaOaaaeaacaaIXaGaey4kaSYaaOaaaeaacaaIXaGaey4kaSYaaOaaaeaacaaIXaGaey4kaSYaaOaaaeaacaaIXaGaey4kaSIaaiOlaiaac6cacaGGUaaaleqaaaqabaaabeaaaeqaaaqabaaaaa@4343@
.Alta
aplicatie a numarului
ϕ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dygaaa@37BB@
apare la pentagonal regulat deoarece :
2cos(
π
5
)=ϕ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiGacogacaGGVbGaai4CaiaacIcadaWcaaqaaiabec8aWbqaaiaaiwdaaaGaaiykaiabg2da9iabew9aMbaa@4035@
si
2sin(
π
5
)=
3−ϕ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiGacohacaGGPbGaaiOBaiaacIcadaWcaaqaaiabec8aWbqaaiaaiwdaaaGaaiykaiabg2da9maakaaabaGaaG4maiabgkHiTiabew9aMbWcbeaaaaa@41FF@
.
De
asemenea exista o legatura intre dreptunghiurile de aur si sirul lui Fibonacci deoarece lungimea si latimea
celui de-al
n−lea
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgkHiTiaadYgacaWGLbGaamyyaaaa@3A94@
dreptughi pot fi scrise ca expresii liniare,
unde coeficientii sunt intotdeauna numere Fibonacci.Aceste
dreptunghiuri pot fi inscrise intr-o spirala logaritmica .Spiralele logaritmice
se intalnesc destul de des in natura(carcasa unui melc, coltii unui elefant sau
conurile de pin).Asemenea spirale sunt echiunghiulare, in sensul ca orice
dreapta ce trece prin punctul
(
x
0
,
y
0
)=(
1+3ϕ
5
,
3−ϕ
5
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhadaWgaaWcbaGaaGimaaqabaGccaGGSaGaamyEamaaBaaaleaacaaIWaaabeaakiaacMcacqGH9aqpcaGGOaWaaSaaaeaacaaIXaGaey4kaSIaaG4maiabew9aMbqaaiaaiwdaaaGaaiilamaalaaabaGaaG4maiabgkHiTiabew9aMbqaaiaaiwdaaaGaaiykaaaa@4818@
taie spirala sub un unghi constant.
2.2. Fibonacci si
plantele.
Plantele
nu au cum sa cunoasca numerele lui Fibonacci,
dar se dezvolta in cel mai eficient mod.
a. multe plante au
aranjamentul frunzelor dispus intr-o
secventa Fibonacci in jurul tulpinei;
b.anumite conuri de
pin respecta o dispunere data de numerele lui Fibonacci ;
c.floarea soarelui
are semintele dispuse dupa o secventa Fibonacci;
d.inelele de pe trunchiurile
palmierilor respecta numerele lui Fibonacci;
e.numarul petalelor
florilor este, de cele mai multe ori, un numar al secventei Fibonacci:
e.1.cala are 1
petala;
e.2.euphorbia are 2
petale;
e.3. irisul si
crinul au 3 petale;
e.4.viorelele,
lalelele, trandafirul salbatic si majoritatea florilor au 5 petale;
e.5.margaretele pot
avea 21 de petale sau 34 de petale si exemplele sunt nenumarate;
e.6.florile cu un
numar de petale care nu sunt in secventa Fibonacci
sun rare si considerate speciale.
Concluzia este
realizarea unui optim, a unei eficiente maxime.Daca se urmeaza secventa lui Fibonacci, frunzele unor plante pot fi
dispuse astfel incat sa ocupe un cat mai mic spatiu si sa obtina cat mai mult
soare.
Ideea dispunerii
frunzelor in acest sens pleaca de la considerarea unghiului de aur de 222.5
grade; unghi care impartit la intregul 360 de grade va da ca rezultat numarul
irational 0.61803398…, cunoscut ca ratia
sirului lui Fibonacci.
2.3. Cochilia melcului, furnica si Fibonacci
Designul
cochiliei melcului urmeaza o spirala foarte reusita, o spirala greu de realizat
cu pixul.Studiata in amanunt s-a ajuns la concluzia ca aceasta spirala urmareste
dimensiunile date de secventa lui Fibonacci:
- pe axa pozitiva :1, 2, 5, 13, s.a.m.d
- pe axa negativa :0, 1, 3, 8, s.a.m.d.
Se observa ca
aceste 2 subsiruri combinate dau numerele lui Fibonacci.
Si in acest caz ratiunea
si motivatia pentru aceasta dispunere este simpla : in acest fel cochilia ii
creaza melcului, in interior un maxim de spatiu si de siguranta.
Furnica are corpul impartit
in trei segmente, dupa diviziunea de aur.
2.4. Fibonacci si
corpul uman.
Fata
umana este caracterizata, din punct de vedere estetic prin cateva dimensiuni
principale: distanta intre ochi, dintre gura si ochi si distanta dintre nas si
ochi, dimensiunea gurii.In stiinta esteticii se apreciaza ca fata este cu atat
considerata mai placuta ochiului cu cat aceste dimensiuni respecta secventa
lui Fibonaci
mai bine.
De exemplu raportul
dintre distanta de la linia surasului(unde se unesc buzele) pana la varful
nasului si de la varful nasului pana la baza sa este aproximativ raportul de
aur
Mana
umana are 5 degete(numar din sirul Fibonacci),
fiecare deget avand 3 falange separate prin 2 incheieturi (numere din sirul Fibonacci).Dimensiunile falangelor
sunt:2 cm, 3 cm, 5 cm.In continuarea lor este un os al palmei care are 8 cm.
2.5. Fibonacci, numarul
de aur si arta.
Daca
privim lucrarile unor mari artisti, fie ei pictori, arhitecti, sculptori sau
fotografi, se observa ca multe dintre ele au la baza regula de aur.Conform
acesteia, “pentru ca un intreg impartit in parti inegale sa para frumos,
trebuie sa existe intre partea mica si cea mare acelasi raport ca intre partea
mare si intreg” (Marcus Pollio Vitruvius,
arhitect roman).
Rudolf Arnheim(psiholog,s-a ocupat de psihologia artei) da o explicatie acestui lucru
astfel:”Acest raport este considerat ca
deosebit de satisfacator datorita modului in care imbina unitatea cu varietatea
dinamica.Intregul si partile sunt perfect proportionate, astfel ca intregul
predomina fara sa fie amenintat de o scindare, iar partile isi pastreaza in
acelasi timp o anumita autonomie.”(in “Arta si perceptia vizuala”).
In
pictura a fost folosit mai ales in Renastere, probabil cea mai discutata
utilizare a acestuia fiind in tabloul lui Leonardo
da Vinci, “Mona Lisa”.Capul, ca si restul corpului e compus utilizand
raportul divin, cum ii spunea da Vinci.In
prima jumatate a secolului trecut pictorul Piet
Mondrian utilizeaza in picturile sale “dreptunghiul de aur”, avand raportul
laturilor aproximativ 1.618…De fapt, lucrarile sale sunt alcatuite numai din
asemenea dreptunghiuri.Acest dreptughi este considerat cea mai armonioasa forma
geometrica.Cu toate acestea, rareori este folosit pentru cadraje.Daca se imparte
fiecare latura a cadrului fotografic in 8 parti egale(numar din sirul Fibonacci) si se unesc punctele de pe
laturile opuse corespunzatoare diviziunilor 3 si 5 (numere din sirul Fibonacci) se obtin asa numitele linii
forte ale cadrului.Punctele aflate la intersectia liniilor se numesc puncte
forte.Practic se pot imparti laturile in trei parti egale, rezultatul este
aproximativ acelasi.Se presupune ca subiectul amplasat pe aceste linii sau in
aceste puncte determina o impartire armonioasa a imaginii astfel incat ea nu
este nici simetrica, nici plictisitoare, nici prea dezechilibrata.De exemplu,
doua fotografii de Robert Doisneau,”L’accordioniste”,
1951 si “The cellist”, 1957 si fotografia “Poplar Trees” a lui Minor White in care toate liniile
converg spre un punct forte.Ansel Adams
se impotrivea regulilor, canoanelor.El spunea “asa zisele reguli de fotocompozitie sunt invalide
, irelevante si imateriale; nu exista reguli de compozitie in fotografie, exista
doar fotografii bune.Cei mai multi fotografi incalca regulile fotocompozitiei”.Cu
toate acestea si in
imaginile
lui se observa diviziunea de aur(vezi fotografia “Aspens”, 1958).Asta inseamna
ca desi nu era de accord cu regulile le cunostea foarte bine.Daca fotografia
are valoare cu subiectul in centru, atunci incalcati regula diviziunii de
aur!Subiectul trebuie sa fie in armonie cu celelalte elemente din cadru.Daca
astfel se verifica si diviziunea de aur, este perfect!Toate acestea arata
importanta acestui numar, astfel ca toti marii fotografi au tinut si tin cont
de el in conceperea unei fotografii.
Pana
si in muzica apare acest raport, se presupune ca Bach sau Beethoven au tinut
cont de el in compozitiile lor.
Atunci cand scrieti,
duceti instinctiv linia din mijloc a literii E(la fel si cu A, F, B, R,…) aproximativ
la
2
3
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIYaaabaGaaG4maaaaaaa@377C@
de baza (aproximativ raportul de aur).
Concluzie. Numerele lui Fibonacci sunt
considerate a fi, de fapt, sistemul de numarare al naturii, un mod de masurare
al Divinitatii, o legatura intre matematica si arta.
3. Unele
rezultate referitoare la sirul lui Fibonacci
Consultand
bibliografia enumerata am selectat urmatoarele rezultate:
Numerele lui Fibonacci
f
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaaaaa@37FD@
sunt date de urmatoarea recurenta :
f
0
=0,
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIWaaabeaakiabg2da9iaaicdacaGGSaaaaa@3A3E@
f
1
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIXaaabeaakiabg2da9iaaigdaaaa@3990@
,
f
n+1
=
f
n−1
+
f
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGH9aqpcaWGMbWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiabgUcaRiaadAgadaWgaaWcbaGaamOBaaqabaaaaa@4152@
,
n≥1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgwMiZkaaigdaaaa@3967@
.
Teorema 1. Daca
x
2
=x+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaaGOmaaaakiabg2da9iaadIhacqGHRaWkcaaIXaaaaa@3B83@
, atunci
avem :
x
n
=
f
n
x+
f
n−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaamOBaaaakiabg2da9iaadAgadaWgaaWcbaGaamOBaaqabaGccaWG4bGaey4kaSIaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaaaaa@40C5@
,
∀n≥2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamOBaiabgwMiZkaaikdaaaa@3A38@
.
Demonstratie.Vom demonstra prin
inductie dupa
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E6@
.
Pentru
n=2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2da9iaaikdaaaa@38A8@
relatia este triviala .Presupunem ca
∀n>2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamOBaiabg6da+iaaikdaaaa@397A@
avem
x
n−1
=
f
n−1
x+
f
n−2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaamOBaiabgkHiTiaaigdaaaGccqGH9aqpcaWGMbWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiaadIhacqGHRaWkcaWGMbWaaSbaaSqaaiaad6gacqGHsislcaaIYaaabeaaaaa@4416@
.
Atunci
x
n
=
x
n−1
⋅x=(
f
n−1
x++
f
n−2
)x=
f
n−1
(x+1)+
f
n−2
x=(
f
n−1
+
f
n−2
)x+
f
n−1
=
f
n
x+
f
n−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@751D@
.
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Teorema 2.(Formula lui Binet).Termenul al
n−lea
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgkHiTiaadYgacaWGLbGaamyyaaaa@3A94@
din sirul lui Fibonacci este dat de:
f
n
=
1
5
(
(
1+
5
2
)
n
−
(
1−
5
2
)
n
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakiabg2da9maalaaabaGaaGymaaqaamaakaaabaGaaGynaaWcbeaaaaGccaGGOaGaaiikamaalaaabaGaaGymaiabgUcaRmaakaaabaGaaGynaaWcbeaaaOqaaiaaikdaaaGaaiykamaaCaaaleqabaGaamOBaaaakiabgkHiTiaacIcadaWcaaqaaiaaigdacqGHsisldaGcaaqaaiaaiwdaaSqabaaakeaacaaIYaaaaiaacMcadaahaaWcbeqaaiaad6gaaaGccaGGPaaaaa@48AD@
,
n≥0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgwMiZkaaicdaaaa@3966@
.
Demonstratie.Radacinile ecuatiei
x
2
=x+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaaGOmaaaakiabg2da9iaadIhacqGHRaWkcaaIXaaaaa@3B83@
sunt
ϕ=
1+
5
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dyMaeyypa0ZaaSaaaeaacaaIXaGaey4kaSYaaOaaaeaacaaI1aaaleqaaaGcbaGaaGOmaaaaaaa@3C0E@
si
1−ϕ=
1−
5
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgkHiTiabew9aMjabg2da9maalaaabaGaaGymaiabgkHiTmaakaaabaGaaGynaaWcbeaaaOqaaiaaikdaaaaaaa@3DC1@
.
Din Teorema 1.,
avem
ϕ
n
=ϕ
f
n
+
f
n−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaWbaaSqabeaacaWGUbaaaOGaeyypa0Jaeqy1dyMaamOzamaaBaaaleaacaWGUbaabeaakiabgUcaRiaadAgadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaaaa@425B@
si
(1−ϕ)
n
=(1−ϕ)
f
n
+
f
n−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaigdacqGHsislcqaHvpGzcaGGPaWaaWbaaSqabeaacaWGUbaaaOGaeyypa0JaaiikaiaaigdacqGHsislcqaHvpGzcaGGPaGaamOzamaaBaaaleaacaWGUbaabeaakiabgUcaRiaadAgadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaaaa@485D@
.
In continuare
ϕ
n
−
(1−ϕ)
n
=
5
f
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaWbaaSqabeaacaWGUbaaaOGaeyOeI0IaaiikaiaaigdacqGHsislcqaHvpGzcaGGPaWaaWbaaSqabeaacaWGUbaaaOGaeyypa0ZaaOaaaeaacaaI1aaaleqaaOGaamOzamaaBaaaleaacaWGUbaabeaaaaa@43B9@
, de unde rezulta formula lui Binet.
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Teorema 3.
f
1
+
f
2
+...+
f
n
=
f
n+2
−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadAgadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcaGGUaGaaiOlaiaac6cacqGHRaWkcaWGMbWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0JaamOzamaaBaaaleaacaWGUbGaey4kaSIaaGOmaaqabaGccqGHsislcaaIXaaaaa@46DC@
.
Demonstratie. Avem relatiile:
f
1
=
f
3
−
f
2
,
f
2
=
f
4
−
f
3
,
f
3
=
f
5
−
f
4
,...,
f
n
=
f
n+2
−
f
n+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5CD0@
,
care prin adunare dau
f
1
+
f
2
+
f
3
+...
f
n
=
f
n+2
−
f
2
=
f
n+2
−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadAgadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcaWGMbWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaaiOlaiaac6cacaGGUaGaamOzamaaBaaaleaacaWGUbaabeaakiabg2da9iaadAgadaWgaaWcbaGaamOBaiabgUcaRiaaikdaaeqaaOGaeyOeI0IaamOzamaaBaaaleaacaaIYaaabeaakiabg2da9iaadAgadaWgaaWcbaGaamOBaiabgUcaRiaaikdaaeqaaOGaeyOeI0IaaGymaaaa@503C@
.
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Teorema 4.
f
1
+
f
3
+
f
5
+...+
f
2n−1
=
f
2n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadAgadaWgaaWcbaGaaG4maaqabaGccqGHRaWkcaWGMbWaaSbaaSqaaiaaiwdaaeqaaOGaey4kaSIaaiOlaiaac6cacaGGUaGaey4kaSIaamOzamaaBaaaleaacaaIYaGaamOBaiabgkHiTiaaigdaaeqaaOGaeyypa0JaamOzamaaBaaaleaacaaIYaGaamOBaaqabaaaaa@496F@
.
Demonstratie. Observam ca:
f
1
=
f
2
−
f
0
,
f
3
=
f
4
−
f
2
,
f
5
=
f
6
−
f
4
,...,
f
2n−1
=
f
2n
−
f
2n−2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIXaaabeaakiabg2da9iaadAgadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWGMbWaaSbaaSqaaiaaicdaaeqaaOGaaiilaiaadAgadaWgaaWcbaGaaG4maaqabaGccqGH9aqpcaWGMbWaaSbaaSqaaiaaisdaaeqaaOGaeyOeI0IaamOzamaaBaaaleaacaaIYaaabeaakiaacYcacaWGMbWaaSbaaSqaaiaaiwdaaeqaaOGaeyypa0JaamOzamaaBaaaleaacaaI2aaabeaakiabgkHiTiaadAgadaWgaaWcbaGaaGinaaqabaGccaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaadAgadaWgaaWcbaGaaGOmaiaad6gacqGHsislcaaIXaaabeaakiabg2da9iaadAgadaWgaaWcbaGaaGOmaiaad6gaaeqaaOGaeyOeI0IaamOzamaaBaaaleaacaaIYaGaamOBaiabgkHiTiaaikdaaeqaaaaa@5F1A@
.Adunam
relatiile si obtinem identitatea dorita.
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Teorema 5.
f
1
2
+
f
2
2
+
f
3
2
+...+
f
n
2
=
f
n
f
n+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIXaaabeaakmaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadAgadaWgaaWcbaGaaGOmaaqabaGcdaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGMbWaaSbaaSqaaiaaiodaaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaiOlaiaac6cacaGGUaGaey4kaSIaamOzamaaBaaaleaacaWGUbaabeaakmaaCaaaleqabaGaaGOmaaaakiabg2da9iaadAgadaWgaaWcbaGaamOBaaqabaGccaWGMbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaaaaa@4DC9@
.
Demonstratie. Avem
f
n−1
f
n
=(
f
n+1
−
f
n
)(
f
n
+
f
n−1
)=
f
n+1
f
n
−
f
n
2
+
f
n+1
f
n−1
−
f
n
f
n−1
.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@675C@
Atunci , obtinem
relatiile
f
n+1
f
n
−
f
n
f
n−1
=
f
n
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccaWGMbWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0IaamOzamaaBaaaleaacaWGUbaabeaakiaadAgadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOGaeyypa0JaamOzamaaBaaaleaacaWGUbaabeaakmaaCaaaleqabaGaaGOmaaaaaaa@4678@
,
care prin adunare pentru
n=1,2,3,...
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacaaIZaGaaiilaiaac6cacaGGUaGaaiOlaaaa@3E46@
,
dau relatia finala.
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Teorema 6. (Identitatea lui Cassini).
f
n−1
f
n+1
−
f
n
2
=
(−1)
n
,n≥1.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccaWGMbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaakiabgkHiTiaadAgadaWgaaWcbaGaamOBaaqabaGcdaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaGGOaGaeyOeI0IaaGymaiaacMcadaahaaWcbeqaaiaad6gaaaGccaGGSaGaamOBaiabgwMiZkaaigdacaGGUaaaaa@4B5B@
Demonstratie.Observam ca :
f
n−1
f
n
−
f
n
2
=(
f
n
−
f
n−2
)(
f
n
+
f
n−1
)−
f
n
2
=−
f
n−2
f
n
−
f
n−1
(
f
n−2
−
f
n
)=−(
f
n−2
f
n
−
f
n−1
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7810@
Daca notam
u
n
=
f
n−1
f
n+1
−
f
n
2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBaaaleaacaWGUbaabeaakiabg2da9iaadAgadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOGaamOzamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGHsislcaWGMbWaaSbaaSqaaiaad6gaaeqaaOWaaWbaaSqabeaacaaIYaaaaaaa@4473@
,
obtinem
u
n
=−
u
n−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBaaaleaacaWGUbaabeaakiabg2da9iabgkHiTiaadwhadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaaaa@3DCA@
si mai departe
u
n
=
(−1)
n−1
u
1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBaaaleaacaWGUbaabeaakiabg2da9iaacIcacqGHsislcaaIXaGaaiykamaaCaaaleqabaGaamOBaiabgkHiTiaaigdaaaGccaWG1bWaaSbaaSqaaiaaigdaaeqaaaaa@40D0@
.
Din cele de mai sus
avem
f
n−1
f
n+1
−
f
n
2
=
(−1)
n−1
(
f
0
f
2
−
f
1
2
)=
(−1)
n
.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccaWGMbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaakiabgkHiTiaadAgadaWgaaWcbaGaamOBaaqabaGcdaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaGGOaGaeyOeI0IaaGymaiaacMcadaahaaWcbeqaaiaad6gacqGHsislcaaIXaaaaOGaaiikaiaadAgadaWgaaWcbaGaaGimaaqabaGccaWGMbWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0IaamOzamaaBaaaleaacaaIXaaabeaakmaaCaaaleqabaGaaGOmaaaakiaacMcacqGH9aqpcaGGOaGaeyOeI0IaaGymaiaacMcadaahaaWcbeqaaiaad6gaaaGccaGGUaaaaa@56DD@
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Teorema 7.(Cesàro).
∑
k=0
n
(
n
k
)
2
k
f
k
=
f
3n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaadaqadaabaeqabaGaamOBaaqaaiaadUgaaaGaayjkaiaawMcaaaWcbaGaam4Aaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaaIYaWaaWbaaSqabeaacaWGRbaaaOGaamOzamaaBaaaleaacaWGRbaabeaakiabg2da9iaadAgadaWgaaWcbaGaaG4maiaad6gaaeqaaaaa@4716@
.
Demonstratie.Utilizam formula
lui Binet ,
∑
k=0
n
(
n
k
)
2
k
f
k
=
∑
k=0
n
(
n
k
)
2
k
ϕ
k
−
(1−ϕ)
k
5
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5A51@
=
=
1
5
(
∑
k
n
(
n
k
)
2
k
ϕ
k
−
∑
k=0
n
(
n
k
)
2
k
(1−ϕ)
k
)
=
1
5
(
(1+2ϕ)
n
−
(1+2(1−ϕ))
n
).
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6E78@
Cum
ϕ
2
=ϕ+1,
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaWbaaSqabeaacaaIYaaaaOGaeyypa0Jaeqy1dyMaey4kaSIaaGymaiaacYcaaaa@3DC9@
obtinem
1+2ϕ=
ϕ
3
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgUcaRiaaikdacqaHvpGzcqGH9aqpcqaHvpGzdaahaaWcbeqaaiaaiodaaaaaaa@3DCC@
si similar
1+2(1−ϕ)=
(1−ϕ)
3
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgUcaRiaaikdacaGGOaGaaGymaiabgkHiTiabew9aMjaacMcacqGH9aqpcaGGOaGaaGymaiabgkHiTiabew9aMjaacMcadaahaaWcbeqaaiaaiodaaaaaaa@43CE@
.
Atunci , rezulta
∑
k=0
n
(
n
k
)
2
k
f
k
=
1
5
(
(ϕ)
3n
+
(1−ϕ)
3n
)=
ϕ
3n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaadaqadaabaeqabaGaamOBaaqaaiaadUgaaaGaayjkaiaawMcaaiaaikdadaahaaWcbeqaaiaadUgaaaGccaWGMbWaaSbaaSqaaiaadUgaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaWaaOaaaeaacaaI1aaaleqaaaaakiaacIcacaGGOaGaeqy1dyMaaiykamaaCaaaleqabaGaaG4maiaad6gaaaGccqGHRaWkcaGGOaGaaGymaiabgkHiTiabew9aMjaacMcadaahaaWcbeqaaiaaiodacaWGUbaaaOGaaiykaiabg2da9iabew9aMnaaBaaaleaacaaIZaGaamOBaaqabaaabaGaam4Aaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdaaaa@5886@
.
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Teorema 8.(Vorobyov)Daca
s≥1,t≥0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiabgwMiZkaaigdacaGGSaGaamiDaiabgwMiZkaaicdaaaa@3D95@
sunt intregi atunci:
f
s+t
=
f
s−1
f
t
+
f
s
f
t+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGZbGaey4kaSIaamiDaaqabaGccqGH9aqpcaWGMbWaaSbaaSqaaiaadohacqGHsislcaaIXaaabeaakiaadAgadaWgaaWcbaGaamiDaaqabaGccqGHRaWkcaWGMbWaaSbaaSqaaiaadohaaeqaaOGaamOzamaaBaaaleaacaWG0bGaey4kaSIaaGymaaqabaaaaa@4770@
.
Demonstratie. Fixam pe
t
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36EC@
si demonstram prin inductie dupa
s
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@36EB@
.Pentru
s=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiabg2da9iaaigdaaaa@38AC@
se obtine
f
t+1
=
f
0
f
t
+
f
1
f
t+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWG0bGaey4kaSIaaGymaaqabaGccqGH9aqpcaWGMbWaaSbaaSqaaiaaicdaaeqaaOGaamOzamaaBaaaleaacaWG0baabeaakiabgUcaRiaadAgadaWgaaWcbaGaaGymaaqabaGccaWGMbWaaSbaaSqaaiaadshacqGHRaWkcaaIXaaabeaaaaa@4510@
,
care este adevarata(trivial).Presupunem ca
s>1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiabg6da+iaaigdaaaa@38AE@
si ca
f
s−k+t
=
f
s−k−1
f
t
+
f
s−k
f
t+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGZbGaeyOeI0Iaam4AaiabgUcaRiaadshaaeqaaOGaeyypa0JaamOzamaaBaaaleaacaWGZbGaeyOeI0Iaam4AaiabgkHiTiaaigdaaeqaaOGaamOzamaaBaaaleaacaWG0baabeaakiabgUcaRiaadAgadaWgaaWcbaGaam4CaiabgkHiTiaadUgaaeqaaOGaamOzamaaBaaaleaacaWG0bGaey4kaSIaaGymaaqabaaaaa@4D07@
pentru orice
k
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E3@
care satisfac
1≤k≤s−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgsMiJkaadUgacqGHKjYOcaWGZbGaeyOeI0IaaGymaaaa@3DA8@
.
Avem
f
s+t
=
f
s+t−1
+
f
s+t−2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGZbGaey4kaSIaamiDaaqabaGccqGH9aqpcaWGMbWaaSbaaSqaaiaadohacqGHRaWkcaWG0bGaeyOeI0IaaGymaaqabaGccqGHRaWkcaWGMbWaaSbaaSqaaiaadohacqGHRaWkcaWG0bGaeyOeI0IaaGOmaaqabaaaaa@46FE@
(din recurenta
Fibonacci)
=
f
s−1+t
+
f
s−2+t
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGZbGaeyOeI0IaaGymaiabgUcaRiaadshaaeqaaOGaey4kaSIaamOzamaaBaaaleaacaWGZbGaeyOeI0IaaGOmaiabgUcaRiaadshaaeqaaaaa@4204@
(trivial)
=
f
s−2
f
t
+
f
s−1
f
t+1
+
f
s−3
f
t
+
f
s−2
f
t+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGZbGaeyOeI0IaaGOmaaqabaGccaWGMbWaaSbaaSqaaiaadshaaeqaaOGaey4kaSIaamOzamaaBaaaleaacaWGZbGaeyOeI0IaaGymaaqabaGccaWGMbWaaSbaaSqaaiaadshacqGHRaWkcaaIXaaabeaakiabgUcaRiaadAgadaWgaaWcbaGaam4CaiabgkHiTiaaiodaaeqaaOGaamOzamaaBaaaleaacaWG0baabeaakiabgUcaRiaadAgadaWgaaWcbaGaam4CaiabgkHiTiaaikdaaeqaaOGaamOzamaaBaaaleaacaWG0bGaey4kaSIaaGymaaqabaaaaa@5339@
(din presupunerea facuta)
=
f
t
(
f
s−2
+
f
s−3
)+
f
t+1
(
f
s−1
+
f
s−2
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWG0baabeaakiaacIcacaWGMbWaaSbaaSqaaiaadohacqGHsislcaaIYaaabeaakiabgUcaRiaadAgadaWgaaWcbaGaam4CaiabgkHiTiaaiodaaeqaaOGaaiykaiabgUcaRiaadAgadaWgaaWcbaGaamiDaiabgUcaRiaaigdaaeqaaOGaaiikaiaadAgadaWgaaWcbaGaam4CaiabgkHiTiaaigdaaeqaaOGaey4kaSIaamOzamaaBaaaleaacaWGZbGaeyOeI0IaaGOmaaqabaGccaGGPaaaaa@5024@
(prin rearanjarea termenilor)
=
f
t
f
s−1
+
f
t+1
f
s
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWG0baabeaakiaadAgadaWgaaWcbaGaam4CaiabgkHiTiaaigdaaeqaaOGaey4kaSIaamOzamaaBaaaleaacaWG0bGaey4kaSIaaGymaaqabaGccaWGMbWaaSbaaSqaaiaadohaaeqaaaaa@4276@
(din
recurenta Fibonacci).
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Observatia 1. J.R. Silvester indica o
metoda eleganta care furnizeaza identitati pentru termenii sirului
(
f
n
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamOBaaqabaGccaGGPaaaaa@3960@
.
Mai precis, se considera matricea
A=
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2da9aaa@37BF@
(
0
1
1
1
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafaqabeGacaaabaGaaGimaaqaaiaaigdaaeaacaaIXaaabaGaaGymaaaaaiaawIcacaGLPaaaaaa@3A77@
si se constata ca avem
A
n
=(
f
n−1
f
n
f
n
f
n+1
),n=1,2,...
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaCaaaleqabaGaamOBaaaakiabg2da9maabmaabaqbaeqabiGaaaqaaiaadAgadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaaaOqaaiaadAgadaWgaaWcbaGaamOBaaqabaaakeaacaWGMbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaaaaaakiaawIcacaGLPaaacaGGSaGaamOBaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlaiaac6caaaa@4DAD@
.Plecand
de la aceasta observatie si utilizand egalitatile
A
n+m
=
A
n
⋅
A
m
,
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaCaaaleqabaGaamOBaiabgUcaRiaad2gaaaGccqGH9aqpcaWGbbWaaWbaaSqabeaacaWGUbaaaOGaeyyXICTaamyqamaaCaaaleqabaGaamyBaaaakiaacYcaaaa@4196@
n,m=1,2,...
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacYcacaWGTbGaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaac6cacaGGUaGaaiOlaaaa@3E7B@
si
det(
A
n
)=
[
det(A)
]
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciizaiaacwgacaGG0bGaaiikaiaadgeadaahaaWcbeqaaiaad6gaaaGccaGGPaGaeyypa0ZaamWaaeaaciGGKbGaaiyzaiaacshacaGGOaGaamyqaiaacMcaaiaawUfacaGLDbaadaahaaWcbeqaaiaad6gaaaaaaa@4509@
,
rezulta relatia din teorema 6. si de asemenea relatia din teorema 8.
Posibilitatile de a
obtine identitati, folosind ideea de mai sus sunt multiple.Astfel, observam ca
A
n
=
f
n
A+
f
n−1
I
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaCaaaleqabaGaamOBaaaakiabg2da9iaadAgadaWgaaWcbaGaamOBaaqabaGccaWGbbGaey4kaSIaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccaWGjbaaaa@412F@
(cu
I
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaaaa@36C1@
am notat matricea untate), iar pentru
n=2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2da9iaaikdaaaa@38A8@
,
obtinem:
A
2
=A+I
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaCaaaleqabaGaaGOmaaaakiabg2da9iaadgeacqGHRaWkcaWGjbaaaa@3B28@
.De
asemenea avem
A
n+2
=
A
n+1
+
A
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaCaaaleqabaGaamOBaiabgUcaRiaaikdaaaGccqGH9aqpcaWGbbWaaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaakiabgUcaRiaadgeadaahaaWcbeqaaiaad6gaaaaaaa@40DC@
.Enuntate
fiind aceste proprietati ale matricei
A
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36B9@
,
se observa ca puterile acesteia verifica recurente de tip Fibonacci.Deoarece
A⋅I=I⋅A=A
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabgwSixlaadMeacqGH9aqpcaWGjbGaeyyXICTaamyqaiabg2da9iaadgeaaaa@4081@
,
putem aplica formula binomului lui Newton
pentru
A
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36B9@
si
I
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaaaa@36C1@
,
apoi identificand relatiile obtinute pe componente se obtin diferite identitati.
Prezentam mai jos,
fara demonstratie, cateva identitati obtinute prin metoda expusa mai sus:
(1)
f
2n
=
∑
k=0
n
(
n
k
)
f
k
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaigdacaGGPaGaamOzamaaBaaaleaacaaIYaGaamOBaaqabaGccqGH9aqpdaaeWbqaamaabmaaeaqabeaacaWGUbaabaGaam4AaaaacaGLOaGaayzkaaGaamOzamaaBaaaleaacaWGRbaabeaaaeaacaWGRbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoaaaa@4731@
;
(2)
f
2n+l
=
∑
k=0
n
(
n
k
)
f
k+l
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaikdacaGGPaGaamOzamaaBaaaleaacaaIYaGaamOBaiabgUcaRiaadYgaaeqaaOGaeyypa0ZaaabCaeaadaqadaabaeqabaGaamOBaaqaaiaadUgaaaGaayjkaiaawMcaaiaadAgadaWgaaWcbaGaam4AaiabgUcaRiaadYgaaeqaaaqaaiaadUgacqGH9aqpcaaIWaaabaGaamOBaaqdcqGHris5aaaa@4AD8@
;
(3)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaiodacaGGPaaaaa@3809@
f
l
=
∑
k=0
n
(
n
k
)
(−1)
k
f
2n−k+l
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGSbaabeaakiabg2da9maaqahabaWaaeWaaqaabeqaaiaad6gaaeaacaWGRbaaaiaawIcacaGLPaaacaGGOaGaeyOeI0IaaGymaiaacMcadaahaaWcbeqaaiaadUgaaaGccaWGMbWaaSbaaSqaaiaaikdacaWGUbGaeyOeI0Iaam4AaiabgUcaRiaadYgaaeqaaaqaaiaadUgacqGH9aqpcaaIWaaabaGaamOBaaqdcqGHris5aaaa@4CF6@
;
(4)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaisdacaGGPaaaaa@380A@
f
l+n
=
∑
k=0
n
(
n
k
)
(−1)
k
f
2n−2k+l
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGSbGaey4kaSIaamOBaaqabaGccqGH9aqpdaaeWbqaamaabmaaeaqabeaacaWGUbaabaGaam4AaaaacaGLOaGaayzkaaGaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGRbaaaOGaamOzamaaBaaaleaacaaIYaGaamOBaiabgkHiTiaaikdacaWGRbGaey4kaSIaamiBaaqabaaabaGaam4Aaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdaaaa@4F87@
;
(5)
f
3n+l
=
∑
k=0
n
(
n
k
)
2
k
f
k+l
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaiwdacaGGPaGaamOzamaaBaaaleaacaaIZaGaamOBaiabgUcaRiaadYgaaeqaaOGaeyypa0ZaaabCaeaadaqadaabaeqabaGaamOBaaqaaiaadUgaaaGaayjkaiaawMcaaiaaikdadaahaaWcbeqaaiaadUgaaaGccaWGMbWaaSbaaSqaaiaadUgacqGHRaWkcaWGSbaabeaaaeaacaWGRbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoaaaa@4CBF@
(teorema
lui Cesàro generalizata);
(6)
2
n
f
n+l
=
∑
k=0
n
(
n
k
)
(−1)
k
f
3n−3k+l
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaiAdacaGGPaGaaGOmamaaCaaaleqabaGaamOBaaaakiaadAgadaWgaaWcbaGaamOBaiabgUcaRiaadYgaaeqaaOGaeyypa0ZaaabCaeaadaqadaabaeqabaGaamOBaaqaaiaadUgaaaGaayjkaiaawMcaaiaacIcacqGHsislcaaIXaGaaiykamaaCaaaleqabaGaam4AaaaakiaadAgadaWgaaWcbaGaaG4maiaad6gacqGHsislcaaIZaGaam4AaiabgUcaRiaadYgaaeqaaaqaaiaadUgacqGH9aqpcaaIWaaabaGaamOBaaqdcqGHris5aaaa@5388@
;
(7)
f
l
=
∑
k=0
n
2
k
(
n
k
)
(−1)
k
f
3n−2k+l
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaiEdacaGGPaGaamOzamaaBaaaleaacaWGSbaabeaakiabg2da9maaqahabaGaaGOmamaaCaaaleqabaGaam4AaaaakmaabmaaeaqabeaacaWGUbaabaGaam4AaaaacaGLOaGaayzkaaGaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGRbaaaOGaamOzamaaBaaaleaacaaIZaGaamOBaiabgkHiTiaaikdacaWGRbGaey4kaSIaamiBaaqabaaabaGaam4Aaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdaaaa@51B0@
;
(8)
f
nm
=
∑
k=0
m
(
m
k
)
f
n
m−k
f
n−1
k
f
m−k
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaiIdacaGGPaGaamOzamaaBaaaleaacaWGUbGaamyBaaqabaGccqGH9aqpdaaeWbqaamaabmaaeaqabeaacaWGTbaabaGaam4AaaaacaGLOaGaayzkaaaaleaacaWGRbGaeyypa0JaaGimaaqaaiaad2gaa0GaeyyeIuoakiaadAgadaWgaaWcbaGaamOBaaqabaGcdaahaaWcbeqaaiaad2gacqGHsislcaWGRbaaaOGaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaahaaWcbeqaaiaadUgaaaGccaWGMbWaaSbaaSqaaiaad2gacqGHsislcaWGRbaabeaaaaa@535D@
.
Pentru studenti
propun demonstrarea urmatoarelor identitati:
(9)
∑
n=2
∞
1
f
n−1
f
n+1
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaiMdacaGGPaWaaabCaeaadaWcaaqaaiaaigdaaeaacaWGMbWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiaadAgadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaaakiabg2da9iaaigdaaSqaaiaad6gacqGH9aqpcaaIYaaabaGaeyOhIukaniabggHiLdaaaa@4870@
;
(10)
∑
n=1
∞
f
n
f
n+1
f
n+2
=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaigdacaaIWaGaaiykamaaqahabaWaaSaaaeaacaWGMbWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOzamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccaWGMbWaaSbaaSqaaiaad6gacqGHRaWkcaaIYaaabeaaaaaabaGaamOBaiabg2da9iaaigdaaeaacqGHEisPa0GaeyyeIuoakiabg2da9iaaigdaaaa@4A65@
;
(11)
∑
n=0
∞
1
f
2
n
=4−ϕ
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaigdacaaIXaGaaiykamaaqahabaWaaSaaaeaacaaIXaaabaGaamOzamaaBaaaleaacaaIYaWaaWbaaWqabeaacaWGUbaaaaWcbeaaaaGccqGH9aqpcaaI0aGaeyOeI0Iaeqy1dygaleaacaWGUbGaeyypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aaaa@4775@
;
(12)
∑
n=1
∞
arctan
1
f
2n+1
=
π
4
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaaciGGHbGaaiOCaiaacogacaGG0bGaaiyyaiaac6gadaWcaaqaaiaaigdaaeaacaWGMbWaaSbaaSqaaiaaikdacaWGUbGaey4kaSIaaGymaaqabaaaaOGaeyypa0ZaaSaaaeaacqaHapaCaeaacaaI0aaaaaWcbaGaamOBaiabg2da9iaaigdaaeaacqGHEisPa0GaeyyeIuoaaaa@4AB6@
;
(13)
lim
n→∞
f
n
ϕ
n
=
1
5
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaigdacaaIZaGaaiykamaaxababaGaciiBaiaacMgacaGGTbaaleaacaWGUbGaeyOKH4QaeyOhIukabeaakmaalaaabaGaamOzamaaBaaaleaacaWGUbaabeaaaOqaaiabew9aMnaaCaaaleqabaGaamOBaaaaaaGccqGH9aqpdaWcaaqaaiaaigdaaeaadaGcaaqaaiaaiwdaaSqabaaaaaaa@47E9@
;
(14)
lim
n→∞
f
n+r
f
n
=
ϕ
r
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaigdacaaI0aGaaiykamaaxababaGaciiBaiaacMgacaGGTbaaleaacaWGUbGaeyOKH4QaeyOhIukabeaakmaalaaabaGaamOzamaaBaaaleaacaWGUbGaey4kaSIaamOCaaqabaaakeaacaWGMbWaaSbaaSqaaiaad6gaaeqaaaaakiabg2da9iabew9aMnaaCaaaleqabaGaamOCaaaaaaa@4A2C@
.
Observatia 2. Unii autori, au obtinut identitati cu
termenii sirului lui Fibonacci cu
ajutorul determinantilor.Astfel, daca consideram sirul lui Fibonacci : 1, 2, 3, 5,…se observa ca
f
n
=
D
n−1
=|
2
1
0
0
...
...
0
1
2
1
0
...
...
0
1
1
2
1
0
...
0
...
...
...
...
...
...
...
1
1
1
...
...
1
2
|
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6D8E@
este deci dat de un determinant de ordinul
n−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgkHiTiaaigdaaaa@388E@
.
Daca consideram
determinantul de ordinul
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E6@
,
D
n
=|
2
1
0
0
...
...
0
1
2
1
0
...
...
0
1
1
2
1
0
...
0
...
...
...
...
...
...
...
1
1
1
...
...
1
2
|
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@68CC@
pe care il dezvoltam
dupa elementele primei linii ne da :
D
n
=2
D
n−1
−|
1
1
0
0
...
...
0
1
2
1
0
...
...
0
1
1
2
1
0
...
0
...
...
...
...
...
...
...
1
1
1
...
...
1
2
|
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6E0E@
.Daca
dezvoltam si ultimul termen obtinem
D
n
=2
D
n−1
−
D
n−2
+|
1
1
0
0
...
...
0
1
2
1
0
...
...
0
1
1
2
1
0
...
0
...
...
...
...
...
...
...
1
1
1
...
...
1
2
|
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@728B@
.Procedam
ca mai sus si deducem :
D
n−1
=2
D
n−2
−|
1
1
0
0
...
...
0
1
2
1
0
...
...
0
1
1
2
1
0
...
0
...
...
...
...
...
...
...
1
1
1
...
...
1
2
|
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGH9aqpcaaIYaGaamiramaaBaaaleaacaWGUbGaeyOeI0IaaGOmaaqabaGccqGHsisldaabdaqaauaabeqafCaaaaaabaGaaGymaaqaaiaaigdaaeaacaaIWaaabaGaaGimaaqaaiaac6cacaGGUaGaaiOlaaqaaiaac6cacaGGUaGaaiOlaaqaaiaaicdaaeaacaaIXaaabaGaaGOmaaqaaiaaigdaaeaacaaIWaaabaGaaiOlaiaac6cacaGGUaaabaGaaiOlaiaac6cacaGGUaaabaGaaGimaaqaaiaaigdaaeaacaaIXaaabaGaaGOmaaqaaiaaigdaaeaacaaIWaaabaGaaiOlaiaac6cacaGGUaaabaGaaGimaaqaaiaac6cacaGGUaGaaiOlaaqaaiaac6cacaGGUaGaaiOlaaqaaiaac6cacaGGUaGaaiOlaaqaaiaac6cacaGGUaGaaiOlaaqaaiaac6cacaGGUaGaaiOlaaqaaiaac6cacaGGUaGaaiOlaaqaaiaac6cacaGGUaGaaiOlaaqaaiaaigdaaeaacaaIXaaabaGaaGymaaqaaiaac6cacaGGUaGaaiOlaaqaaiaac6cacaGGUaGaaiOlaaqaaiaaigdaaeaacaaIYaaaaaGaay5bSlaawIa7aaaa@6FB7@
.Daca
adunam membru cu membru ultimile doua egalitati obtinem :
D
n
+
D
n−1
=2
D
n−1
−
D
n−2
+2
D
n−2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBaaaleaacaWGUbaabeaakiabgUcaRiaadseadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOGaeyypa0JaaGOmaiaadseadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOGaeyOeI0IaamiramaaBaaaleaacaWGUbGaeyOeI0IaaGOmaaqabaGccqGHRaWkcaaIYaGaamiramaaBaaaleaacaWGUbGaeyOeI0IaaGOmaaqabaaaaa@4B74@
⇔
D
n
=
D
n−1
+
D
n−2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSTaamiramaaBaaaleaacaWGUbaabeaakiabg2da9iaadseadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOGaey4kaSIaamiramaaBaaaleaacaWGUbGaeyOeI0IaaGOmaaqabaaaaa@4354@
,
relatie de recurenta
analoaga cu cea din sirul lui Fibonacci
f
n+1
=
f
n−1
+
f
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGH9aqpcaWGMbWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiabgUcaRiaadAgadaWgaaWcbaGaamOBaaqabaaaaa@4152@
.
Teorema 9.
f
n
|
f
nk
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakmaaeeaabaGaamOzamaaBaaaleaacaWGUbGaam4AaaqabaaakiaawEa7aaaa@3C9F@
,
∀n,k∈
N
∗
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamOBaiaacYcacaWGRbGaeyicI4SaamOtamaaCaaaleqabaGaey4fIOcaaaaa@3CC9@
.
Demonstratie. Prin inductie dupa
k∈
N
∗
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgIGiolaad6eadaahaaWcbeqaaiabgEHiQaaaaaa@3A56@
si orice
n∈
N
∗
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolaad6eadaahaaWcbeqaaiabgEHiQaaaaaa@3A59@
.
Pentru
k=1⇒
f
n
=
f
n⋅1
⇒
f
n
|
f
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2da9iaaigdacqGHshI3caWGMbWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0JaamOzamaaBaaaleaacaWGUbGaeyyXICTaaGymaaqabaGccqGHshI3caWGMbWaaSbaaSqaaiaad6gaaeqaaOWaaqqaaeaacaWGMbWaaSbaaSqaaiaad6gaaeqaaaGccaGLhWoaaaa@4B4D@
(adevarata).
Presupunem
f
n
|
f
nk
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakmaaeeaabaGaamOzamaaBaaaleaacaWGUbGaam4AaaqabaaakiaawEa7aaaa@3C9F@
si demonstram ca
f
n
|
f
n(k+1)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakmaaeeaabaGaamOzamaaBaaaleaacaWGUbGaaiikaiaadUgacqGHRaWkcaaIXaGaaiykaaqabaaakiaawEa7aaaa@3F95@
.
Intradevar, tinand
seama de teorema 8. avem:
f
n(k+1)
=
f
nk+n
=
f
nk+1
f
n
+
f
nk
f
n−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbGaaiikaiaadUgacqGHRaWkcaaIXaGaaiykaaqabaGccqGH9aqpcaWGMbWaaSbaaSqaaiaad6gacaWGRbGaey4kaSIaamOBaaqabaGccqGH9aqpcaWGMbWaaSbaaSqaaiaad6gacaWGRbGaey4kaSIaaGymaaqabaGccaWGMbWaaSbaaSqaaiaad6gaaeqaaOGaey4kaSIaamOzamaaBaaaleaacaWGUbGaam4AaaqabaGccaWGMbWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaaaaa@511F@
,
∀n∈
N
∗
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamOBaiabgIGiolaad6eadaahaaWcbeqaaiabgEHiQaaaaaa@3B29@
si deoarece
f
n
|
f
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakmaaeeaabaGaamOzamaaBaaaleaacaWGUbaabeaaaOGaay5bSdaaaa@3BAF@
si
f
|
f
nk
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBeaaleaacaWGUbaabeaakmaaeeaabaGaamOzamaaBaaaleaacaWGUbGaam4AaaqabaaakiaawEa7aaaa@3CA0@
rezulta
f
n
|
f
n(k+1)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakmaaeeaabaGaamOzamaaBaaaleaacaWGUbGaaiikaiaadUgacqGHRaWkcaaIXaGaaiykaaqabaaakiaawEa7aaaa@3F95@
.
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Teorema 10.
f
kn−1
≡
f
n−1
k
(mod
f
n
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGRbGaamOBaiabgkHiTiaaigdaaeqaaOGaeyyyIORaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaahaaWcbeqaaiaadUgaaaGccaGGOaGaciyBaiaac+gacaGGKbGaamOzamaaBaaaleaacaWGUbaabeaakmaaCaaaleqabaGaaGOmaaaakiaacMcaaaa@4879@
,
∀k,n∈
N
∗
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaam4AaiaacYcacaWGUbGaeyicI4SaamOtamaaCaaaleqabaGaey4fIOcaaaaa@3CC9@
.
Demonstratie.Se arata tot prin
inductie dupa
k
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E3@
∈
N
∗
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyicI4SaamOtamaaCaaaleqabaGaey4fIOcaaaaa@3966@
.Pentru
k=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2da9iaaigdaaaa@38A4@
relatia este evidenta.Pentru
k=2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2da9iaaikdaaaa@38A5@
tinem seama de teoremele precedente si avem ;
f
2n−1
=
f
n
f
n
+
f
n−1
f
n−1
≡
f
n−1
2
(mod
f
n
2
),∀n∈
N
∗
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaaIYaGaamOBaiabgkHiTiaaigdaaeqaaOGaeyypa0JaamOzamaaBaaaleaacaWGUbaabeaakiaadAgadaWgaaWcbaGaamOBaaqabaGccqGHRaWkcaWGMbWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiaadAgadaWgaaWcbaGaamOBaiabgkHiTiaaigdaaeqaaOGaeyyyIORaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaahaaWcbeqaaiaaikdaaaGccaGGOaGaciyBaiaac+gacaGGKbGaamOzamaaBaaaleaacaWGUbaabeaakmaaCaaaleqabaGaaGOmaaaakiaacMcacaGGSaGaeyiaIiIaamOBaiabgIGiolaad6eadaahaaWcbeqaaiabgEHiQaaaaaa@5B7F@
.Fie
f
kn−1
≡
f
n−1
k
(mod
f
n
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGRbGaamOBaiabgkHiTiaaigdaaeqaaOGaeyyyIORaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaahaaWcbeqaaiaadUgaaaGccaGGOaGaciyBaiaac+gacaGGKbGaamOzamaaBaaaleaacaWGUbaabeaakmaaCaaaleqabaGaaGOmaaaakiaacMcaaaa@4879@
.Avem
deci ca :
f
(k+1)n−1
=
f
kn−1+n
=
f
kn
f
n
+
f
kn−1
f
n−1
≡
f
kn−1
f
n−1
≡
f
n−1
k
f
n−1
≡
f
n−1
k+1
(mod
f
n
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7894@
si deci conform principiului inductiei
complete relatia este demonstrata.
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Teorema 11.
f
kn−1
≡
(−1)
k+1
f
n−2
k
(mod
f
n
2
),∀n∈
N
∗
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGRbGaamOBaiabgkHiTiaaigdaaeqaaOGaeyyyIORaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGRbGaey4kaSIaaGymaaaakiaadAgadaWgaaWcbaGaamOBaiabgkHiTiaaikdaaeqaaOWaaWbaaSqabeaacaWGRbaaaOGaaiikaiGac2gacaGGVbGaaiizaiaadAgadaWgaaWcbaGaamOBaaqabaGcdaahaaWcbeqaaiaaikdaaaGccaGGPaGaaiilaiabgcGiIiaad6gacqGHiiIZcaWGobWaaWbaaSqabeaacqGHxiIkaaaaaa@5425@
.
Demonstratie.Relatia se
demonstreaza tot prin inductie completa .Pentru
k=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2da9iaaigdaaaa@38A4@
obtinem
f
n−2
≡
f
n−2
(mod
f
n
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGOmaaqabaGccqGHHjIUcaWGMbWaaSbaaSqaaiaad6gacqGHsislcaaIYaaabeaakiaacIcaciGGTbGaai4BaiaacsgacaWGMbWaaSbaaSqaaiaad6gaaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaaiykaaaa@4664@
ceea ce este evident.Presupunem ca
f
kn−2
≡
(−1)
k+1
f
n−2
k
(mod
f
n
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGRbGaamOBaiabgkHiTiaaikdaaeqaaOGaeyyyIORaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGRbGaey4kaSIaaGymaaaakiaadAgadaWgaaWcbaGaamOBaiabgkHiTiaaikdaaeqaaOWaaWbaaSqabeaacaWGRbaaaOGaaiikaiGac2gacaGGVbGaaiizaiaadAgadaWgaaWcbaGaamOBaaqabaGcdaahaaWcbeqaaiaaikdaaaGccaGGPaaaaa@4E40@
si sa demonstram ca
f
(k+1)n−2
≡
(−1)
k+2
f
n−2
k+1
(mod
f
n
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaGGOaGaam4AaiabgUcaRiaaigdacaGGPaGaamOBaiabgkHiTiaaikdaaeqaaOGaeyyyIORaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGRbGaey4kaSIaaGOmaaaakiaadAgadaWgaaWcbaGaamOBaiabgkHiTiaaikdaaeqaaOWaaWbaaSqabeaacaWGRbGaey4kaSIaaGymaaaakiaacIcaciGGTbGaai4BaiaacsgacaWGMbWaaSbaaSqaaiaad6gaaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaaiykaaaa@52D4@
.Intradevar, avem:
f
(k+1)n−2
=
f
kn−2+n
=
f
kn−1
f
n
+
f
kn−2
f
n−1
≡
f
kn−1
f
n
+
(−1)
k+1
f
n−2
k
(
f
n
−
f
n−2
)≡
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7423@
≡
f
kn−1
f
n
+
(−1)
k+1
f
n−2
k
f
n
+
(−1)
k+2
f
n−2
k+1
≡(
f
n−1
k
+
(−1)
k+1
f
n−2
k
)
f
n
+
(−1)
k+2
f
n−2
k+1
≡
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7DDA@
≡
(−1)
k+2
f
n−2
k+1
(mod
f
n
2
).
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyyIORaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGRbGaey4kaSIaaGOmaaaakiaadAgadaWgaaWcbaGaamOBaiabgkHiTiaaikdaaeqaaOWaaWbaaSqabeaacaWGRbGaey4kaSIaaGymaaaakiaacIcaciGGTbGaai4BaiaacsgacaWGMbWaaSbaaSqaaiaad6gaaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaaiykaiaac6caaaa@4BE3@
Am folosit mai sus
ca
f
n−1
k
+
(−1)
k+1
f
n−2
k
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGcdaahaaWcbeqaaiaadUgaaaGccqGHRaWkcaGGOaGaeyOeI0IaaGymaiaacMcadaahaaWcbeqaaiaadUgacqGHRaWkcaaIXaaaaOGaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGOmaaqabaGcdaahaaWcbeqaaiaadUgaaaaaaa@4657@
se divide cu
f
n−1
+
f
n−2
=
f
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGHRaWkcaWGMbWaaSbaaSqaaiaad6gacqGHsislcaaIYaaabeaakiabg2da9iaadAgadaWgaaWcbaGaamOBaaqabaaaaa@415E@
.
Rezulta conform
principiului inductiei complete ca teorema este demonstrata.
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Teorema 12.
f
n
2
|
f
n
f
n
,∀n∈
N
∗
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakmaaCaaaleqabaGaaGOmaaaakmaaeeaabaGaamOzamaaBaaaleaacaWGUbGaamOzamaaBaaameaacaWGUbaabeaaaSqabaaakiaawEa7aiaacYcacqGHaiIicaWGUbGaeyicI4SaamOtamaaCaaaleqabaGaey4fIOcaaaaa@449E@
.
Demonstratie.Notam
f
n
=k
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakiabg2da9iaadUgaaaa@39FD@
.Avem:
f
n
f
n
=
f
nk
=
f
nk−1
+
f
nk−2
≡
f
n−1
k
+
(−1)
k+1
f
n−2
k
(mod
k
2
).
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@625B@
Totodata
avem:
f
n−1
=
f
n
−
f
n−2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccqGH9aqpcaWGMbWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0IaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGOmaaqabaaaaa@4169@
si deci
f
n−1
k
=
(
f
n
−
f
n−2
)
k
=
∑
i=0
k
(
k
i
)
(−1)
i
f
n
k−i
f
n−2
i
≡
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5F4A@
(−1)
k
f
n−2
k
(mod
k
2
).
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGRbaaaOGaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGOmaaqabaGcdaahaaWcbeqaaiaadUgaaaGccaGGOaGaciyBaiaac+gacaGGKbGaam4AamaaCaaaleqabaGaaGOmaaaakiaacMcacaGGUaaaaa@45BB@
Deci,
f
n−1
k
+
(−1)
k+1
f
n−2
k
≡
(−1)
k
f
n−2
k
+
(−1)
k+1
f
n−2
k
≡0(mod
k
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@654E@
,
ceea ce demonstreaza teorema.
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Teorema 13.
f
n
m+1
|
f
n
f
n
m
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakmaaCaaaleqabaGaamyBaiabgUcaRiaaigdaaaGcdaabbaqaaiaadAgadaWgaaWcbaGaamOBaiaadAgadaWgaaadbaGaamOBaaqabaWcdaahaaadbeqaaiaad2gaaaaaleqaaaGccaGLhWoaaaa@41B6@
,
∀m∈N
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamyBaiabgIGiolaad6eaaaa@3A0C@
si
∀n∈
N
∗
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamOBaiabgIGiolaad6eadaahaaWcbeqaaiabgEHiQaaaaaa@3B29@
.
Demonstratie.Pentru
m=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg2da9iaaigdaaaa@38A6@
afirmatia este echivalenta cu teorema
12.Presupunem afirmatia adevarata pentru pentru
m
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36E5@
si o demonstram pentru
m+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabgUcaRiaaigdaaaa@3882@
.Vom
arata ca
f
n
m+1
|
f
n
f
n
m
⇒
f
n
m+2
|
f
n
f
n
m+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakmaaCaaaleqabaGaamyBaiabgUcaRiaaigdaaaGcdaabbaqaaiaadAgadaWgaaWcbaGaamOBaiaadAgadaWgaaadbaGaamOBaaqabaWcdaahaaadbeqaaiaad2gaaaaaleqaaaGccaGLhWoacqGHshI3caWGMbWaaSbaaSqaaiaad6gaaeqaaOWaaWbaaSqabeaacaWGTbGaey4kaSIaaGOmaaaakmaaeeaabaGaamOzamaaBaaaleaacaWGUbGaamOzamaaBaaameaacaWGUbaabeaalmaaCaaameqabaGaamyBaiabgUcaRiaaigdaaaaaleqaaaGccaGLhWoaaaa@5174@
.Notam
si atunci avem:
f
u
f
n
=
f
u
f
n
−1
+
f
u
f
n
−2
≡
f
u−1
f
n
+
(−1)
f
n
+1
f
u−2
f
n
(mod
u
2
)≡
(
f
u
−
f
u−2
)
f
n
+
(−1)
f
n
+1
f
u−2
f
n
(mod
u
2
)≡
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8554@
≡
(−1)
f
n
f
u−2
f
n
+
(−1)
f
n
+1
f
u−2
f
n
≡0(mod
f
n
m+1
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5C5C@
.Am
folosit mai sus ca din
f
n
m
|u
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakmaaCaaaleqabaGaamyBaaaakmaaeeaabaGaamyDaaGaay5bSdaaaa@3BBE@
si
f
n
m
|
u
2
⇒
f
n
m+1
|
f
u
.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakmaaCaaaleqabaGaamyBaaaakmaaeeaabaGaamyDamaaCaaaleqabaGaaGOmaaaaaOGaay5bSdGaeyO0H4TaamOzamaaBaaaleaacaWGUbaabeaakmaaCaaaleqabaGaamyBaiabgUcaRiaaigdaaaGcdaabbaqaaiaadAgadaWgaaWcbaGaamyDaaqabaaakiaawEa7aiaac6caaaa@4849@
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Teorema 14.Orice doua numere Fibonacci consecutive sunt relative
prime.
Demonstratie.Fie
d=(
f
n
,
f
n+1
).
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabg2da9iaacIcacaWGMbWaaSbaaSqaaiaad6gaaeqaaOGaaiilaiaadAgadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaOGaaiykaiaac6caaaa@4062@
Avem
f
n+1
−
f
n
=
f
n−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGHsislcaWGMbWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0JaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaaaaa@415D@
de unde rezulta
d|
f
n−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaeeaabaGaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaaakiaawEa7aaaa@3C2C@
.Atunci
d|
(
f
n
−
f
n−1
)=
f
n−2
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaeeaabaGaaiikaiaadAgadaWgaaWcbaGaamOBaaqabaGccqGHsislcaWGMbWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiaacMcacqGH9aqpcaWGMbWaaSbaaSqaaiaad6gacqGHsislcaaIYaaabeaaaOGaay5bSdaaaa@4549@
.Repetand
procedeul se deduce ca
d|
f
1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaeeaabaGaamOzamaaBaaaleaacaaIXaaabeaaaOGaay5bSdaaaa@3A4C@
,
deci
d=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabg2da9iaaigdaaaa@389D@
.
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Altfel:din teorema
6.,
f
n−1
f
n+1
−
f
n
2
=
(−1)
n
.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaGccaWGMbWaaSbaaSqaaiaad6gacqGHRaWkcaaIXaaabeaakiabgkHiTiaadAgadaWgaaWcbaGaamOBaaqabaGcdaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaGGOaGaeyOeI0IaaGymaiaacMcadaahaaWcbeqaaiaad6gaaaGccaGGUaaaaa@4737@
Rezulta
d|
(−1)
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaeeaabaGaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGUbaaaaGccaGLhWoaaaa@3C9B@
,
i.e.,
d=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabg2da9iaaigdaaaa@389D@
.
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Teorema 15.
(
f
m
,
f
n
)=
f
(m,n)
.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamyBaaqabaGccaGGSaGaamOzamaaBaaaleaacaWGUbaabeaakiaacMcacqGH9aqpcaWGMbWaaSbaaSqaaiaacIcacaWGTbGaaiilaiaad6gacaGGPaaabeaakiaac6caaaa@42EA@
Demonstratie.Notam
a=(m,n),b=(
f
m
,
f
n
),c=
f
(m,n)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2da9iaacIcacaWGTbGaaiilaiaad6gacaGGPaGaaiilaiaadkgacqGH9aqpcaGGOaGaamOzamaaBaaaleaacaWGTbaabeaakiaacYcacaWGMbWaaSbaaSqaaiaad6gaaeqaaOGaaiykaiaacYcacaWGJbGaeyypa0JaamOzamaaBaaaleaacaGGOaGaamyBaiaacYcacaWGUbGaaiykaaqabaaaaa@4C3D@
.Vom
arata ca
c|b
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaeeaabaGaamOyaaGaay5bSdaaaa@3956@
si
b|c
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaeeaabaGaam4yaaGaay5bSdaaaa@3956@
.
Deoarece
a|m
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaeeaabaGaamyBaaGaay5bSdaaaa@395F@
si
a|n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaeeaabaGaamOBaaGaay5bSdaaaa@3960@
,
conform teoremei 9. avem:
f
a
|
f
m
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGHbaabeaakmaaeeaabaGaamOzamaaBaaaleaacaWGTbaabeaaaOGaay5bSdaaaa@3BA1@
si
f
a
|
f
n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGHbaabeaakmaaeeaabaGaamOzamaaBaaaleaacaWGUbaabeaaaOGaay5bSdaaaa@3BA2@
.Deci
f
a
|
(
f
m
,
f
n
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGHbaabeaakmaaeeaabaGaaiikaiaadAgadaWgaaWcbaGaamyBaaqabaGccaGGSaGaamOzamaaBaaaleaacaWGUbaabeaakiaacMcaaiaawEa7aaaa@3FBE@
,
i.e.,
c|b
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaeeaabaGaamOyaaGaay5bSdaaaa@3956@
.
Acum, din Teorema Bachet-Bezout, exista numerele intregi
x,y
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaacYcacaWG5baaaa@389E@
astfel incat
xm+yn=a
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaad2gacqGHRaWkcaWG5bGaamOBaiabg2da9iaadggaaaa@3CA1@
.Se
observa ca
x
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F0@
si
y
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36F1@
nu pot fi ambele negative, deoarece
a
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36D9@
ar fi negativ.Cum
a|n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaeeaabaGaamOBaaGaay5bSdaaaa@3960@
si
a|m
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaeeaabaGaamyBaaGaay5bSdaaaa@395F@
,
avem
a≤n,a≤m
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgsMiJkaad6gacaGGSaGaamyyaiabgsMiJkaad2gaaaa@3DBE@
.De
asemenea,
x
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F0@
si
y
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36F1@
nu pot fi simultan pozitive , deoarece am avea
a=xm+yn≥m+n
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2da9iaadIhacaWGTbGaey4kaSIaamyEaiaad6gacqGHLjYScaWGTbGaey4kaSIaamOBaaaa@412E@
,
contradictie.Atunci ,
x
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F0@
si
y
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36F1@
au semne diferite si, fara a restrange
generalitatea presupunem ca
x≤0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgsMiJkaaicdaaaa@395F@
,
y>0
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabg6da+iaaicdaaaa@38B3@
.
Observam ca :
f
yn
=
f
a−xm
=
f
a−1
f
−xm
+
f
a
f
−xm+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWG5bGaamOBaaqabaGccqGH9aqpcaWGMbWaaSbaaSqaaiaadggacqGHsislcaWG4bGaamyBaaqabaGccqGH9aqpcaWGMbWaaSbaaSqaaiaadggacqGHsislcaaIXaaabeaakiaadAgadaWgaaWcbaGaeyOeI0IaamiEaiaad2gaaeqaaOGaey4kaSIaamOzamaaBaaaleaacaWGHbaabeaakiaadAgadaWgaaWcbaGaeyOeI0IaamiEaiaad2gacqGHRaWkcaaIXaaabeaaaaa@5019@
(am utilizat teorema 8.).
Cum
n|
yn
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaeeaabaGaamyEaiaad6gaaiaawEa7aaaa@3A6B@
si
m|
(−xm)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaeeaabaGaaiikaiabgkHiTiaadIhacaWGTbGaaiykaaGaay5bSdaaaa@3CAE@
,
din teorema 9. rezulta ca
f
n
|
f
yn
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGUbaabeaakmaaeeaabaGaamOzamaaBaaaleaacaWG5bGaamOBaaqabaaakiaawEa7aaaa@3CAD@
si
f
m
|
f
−xm
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBaaaleaacaWGTbaabeaakmaaeeaabaGaamOzamaaBaaaleaacqGHsislcaWG4bGaamyBaaqabaaakiaawEa7aaaa@3D97@
.Acestea
implica ca
(
f
m
,
f
n
)|
f
yn
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamyBaaqabaGccaGGSaGaamOzamaaBaaaleaacaWGUbaabeaakiaacMcadaabbaqaaiaadAgadaWgaaWcbaGaamyEaiaad6gaaeqaaaGccaGLhWoaaaa@40C9@
si
(
f
m
,
f
n
)|
f
−xm
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamyBaaqabaGccaGGSaGaamOzamaaBaaaleaacaWGUbaabeaakiaacMcadaabbaqaaiaadAgadaWgaaWcbaGaeyOeI0IaamiEaiaad2gaaeqaaaGccaGLhWoaaaa@41B4@
.Din
cele de mai sus avem ca
(
f
m
,
f
n
)|
f
a
f
−xm+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamyBaaqabaGccaGGSaGaamOzamaaBaaaleaacaWGUbaabeaakiaacMcadaabbaqaaiaadAgadaWgaaWcbaGaamyyaaqabaGccaWGMbWaaSbaaSqaaiabgkHiTiaadIhacaWGTbGaey4kaSIaaGymaaqabaaakiaawEa7aaaa@4558@
.
Daca
(
f
m
,
f
n
)|
f
−xm+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamyBaaqabaGccaGGSaGaamOzamaaBaaaleaacaWGUbaabeaakiaacMcadaabbaqaaiaadAgadaWgaaWcbaGaeyOeI0IaamiEaiaad2gacqGHRaWkcaaIXaaabeaaaOGaay5bSdaaaa@4351@
,
cum
(
f
m
,
f
n
)|
f
−xm
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamyBaaqabaGccaGGSaGaamOzamaaBaaaleaacaWGUbaabeaakiaacMcadaabbaqaaiaadAgadaWgaaWcbaGaeyOeI0IaamiEaiaad2gaaeqaaaGccaGLhWoaaaa@41B4@
rezulta ca
(
f
m
,
f
n
)
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamyBaaqabaGccaGGSaGaamOzamaaBaaaleaacaWGUbaabeaakiaacMcaaaa@3C23@
ar divide doua numere Fibonacci consecutive, contradictie( conform teoremei 14 ) in cazul
in care
(
f
m
,
f
n
)>1.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamyBaaqabaGccaGGSaGaamOzamaaBaaaleaacaWGUbaabeaakiaacMcacqGH+aGpcaaIXaGaaiOlaaaa@3E98@
Cazul
(
f
m
,
f
n
)=1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamyBaaqabaGccaGGSaGaamOzamaaBaaaleaacaWGUbaabeaakiaacMcacqGH9aqpcaaIXaaaaa@3DE4@
este trivial.Rezulta ca
(
f
m
,
f
n
)|
f
a
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgadaWgaaWcbaGaamyBaaqabaGccaGGSaGaamOzamaaBaaaleaacaWGUbaabeaakiaacMcadaabbaqaaiaadAgadaWgaaWcbaGaamyyaaqabaaakiaawEa7aaaa@3FBE@
,
ceea ce trebuia demonstrat.
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Teorema 16.Daca
p≠5
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabgcMi5kaaiwdaaaa@396E@
este un numar prim impar, atunci
p|
f
p−1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaeeaabaGaamOzamaaBaaaleaacaWGWbGaeyOeI0IaaGymaaqabaaakiaawEa7aaaa@3C3A@
sau
p|
f
p+1
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaeeaabaGaamOzamaaBaaaleaacaWGWbGaey4kaSIaaGymaaqabaaakiaawEa7aaaa@3C2F@
.
Demonstratie.
Lema 1.
(
p−1
n
)≡
(−1)
n
modp,1≤n≤p−1.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaqaabeqaaiaadchacqGHsislcaaIXaaabaGaaeiiaiaabccacaqGGaGaamOBaaaacaGLOaGaayzkaaGaeyyyIORaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGUbaaaOGaciyBaiaac+gacaGGKbGaamiCaiaacYcacaaIXaGaeyizImQaamOBaiabgsMiJkaadchacqGHsislcaaIXaGaaiOlaaaa@4FCA@
Demonstratie.
(p−1)(p−2)...(p−n)≡(−1)(−2)...(−n)≡
(−1)
n
n!modp
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadchacqGHsislcaaIXaGaaiykaiaacIcacaWGWbGaeyOeI0IaaGOmaiaacMcacaGGUaGaaiOlaiaac6cacaGGOaGaamiCaiabgkHiTiaad6gacaGGPaGaeyyyIORaaiikaiabgkHiTiaaigdacaGGPaGaaiikaiabgkHiTiaaikdacaGGPaGaaiOlaiaac6cacaGGUaGaaiikaiabgkHiTiaad6gacaGGPaGaeyyyIORaaiikaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaWGUbaaaOGaamOBaiaacgcaciGGTbGaai4BaiaacsgacaWGWbaaaa@5C8E@
,
de aici concluzia.
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Lema 2.
(
p+1
n
)≡0modp,2≤n≤p−1.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaqaabeqaaiaadchacqGHRaWkcaaIXaaabaGaaeiiaiaabccacaqGGaGaamOBaaaacaGLOaGaayzkaaGaeyyyIORaaGimaiGac2gacaGGVbGaaiizaiaadchacaGGSaGaaGOmaiabgsMiJkaad6gacqGHKjYOcaWGWbGaeyOeI0IaaGymaiaac6caaaa@4C4F@
Demonstratie.
(p+1)(p)(p−1)...(p−n−2)≡(1)(0)(−1)...(−n)≡0modp
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadchacqGHRaWkcaaIXaGaaiykaiaacIcacaWGWbGaaiykaiaacIcacaWGWbGaeyOeI0IaaGymaiaacMcacaGGUaGaaiOlaiaac6cacaGGOaGaamiCaiabgkHiTiaad6gacqGHsislcaaIYaGaaiykaiabggMi6kaacIcacaaIXaGaaiykaiaacIcacaaIWaGaaiykaiaacIcacqGHsislcaaIXaGaaiykaiaac6cacaGGUaGaaiOlaiaacIcacqGHsislcaWGUbGaaiykaiabggMi6kaaicdaciGGTbGaai4BaiaacsgacaWGWbaaaa@5C95@
,ceea
ce demonstreaza lema.
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Din teorema 2.
avem:
f
n
=
1
2
n−1
((
n
1
)+5(
n
3
)+
5
2
(
n
5
)+...+
5
n−2
2
(
n
n−1
))
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5B50@
.
Din lema 1.,
2
p−2
f
p−1
≡p−1−(5+
5
2
+...+
5
p−3
2
)≡−
5
p−1
2
−1
4
modp
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCaaaleqabaGaamiCaiabgkHiTiaaikdaaaGccaWGMbWaaSbaaSqaaiaadchacqGHsislcaaIXaaabeaakiabggMi6kaadchacqGHsislcaaIXaGaeyOeI0IaaiikaiaaiwdacqGHRaWkcaaI1aWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaiOlaiaac6cacaGGUaGaey4kaSIaaGynamaaCaaaleqabaWaaSaaaeaacaWGWbGaeyOeI0IaaG4maaqaaiaaikdaaaaaaOGaaiykaiabggMi6kabgkHiTmaalaaabaGaaGynamaaCaaaleqabaWaaSaaaeaacaWGWbGaeyOeI0IaaGymaaqaaiaaikdaaaaaaOGaeyOeI0IaaGymaaqaaiaaisdaaaGaciyBaiaac+gacaGGKbGaamiCaaaa@5CCA@
.
Din lema 2.,
2
p
f
p+1
≡p+1+
5
p−1
2
≡(
5
p−1
2
+1)modp.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCaaaleqabaGaamiCaaaakiaadAgadaWgaaWcbaGaamiCaiabgUcaRiaaigdaaeqaaOGaeyyyIORaamiCaiabgUcaRiaaigdacqGHRaWkcaaI1aWaaWbaaSqabeaadaWcaaqaaiaadchacqGHsislcaaIXaaabaGaaGOmaaaaaaGccqGHHjIUcaGGOaGaaGynamaaCaaaleqabaWaaSaaaeaacaWGWbGaeyOeI0IaaGymaaqaaiaaikdaaaaaaOGaey4kaSIaaGymaiaacMcaciGGTbGaai4BaiaacsgacaWGWbGaaiOlaaaa@52BD@
Din cele doua relatii
de mai sus obtinem:
2
2p
f
p−1
f
p+1
≡−(
5
p−1
−1)modp
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCaaaleqabaGaaGOmaiaadchaaaGccaWGMbWaaSbaaSqaaiaadchacqGHsislcaaIXaaabeaakiaadAgadaWgaaWcbaGaamiCaiabgUcaRiaaigdaaeqaaOGaeyyyIORaeyOeI0IaaiikaiaaiwdadaahaaWcbeqaaiaadchacqGHsislcaaIXaaaaOGaeyOeI0IaaGymaiaacMcaciGGTbGaai4BaiaacsgacaWGWbaaaa@4D15@
.
Din mica teorema a
lui Fermat,
5
p−1
≡1mod p
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynamaaCaaaleqabaGaamiCaiabgkHiTiaaigdaaaGccqGHHjIUcaaIXaGaaeyBaiaab+gacaqGKbGaaeiiaiaadchaaaa@406B@
,
pentru
p≠5
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabgcMi5kaaiwdaaaa@396E@
si teorema este demonstrata.
�
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbyaqaaaaaaaaaWdbiabl2==Ubaa@39FE@
Nota. Punctul de plecare al
acestui articol l-a constituit raspunsul dat de dl. prof. dr. Ioan Tomescu
(Membru Corespondent al Academiei), Secretarului General al S.S.M.R din Romania
dl. prof. Mircea Trifu, in Gazeta
Matematica nr.12 / 2007, la intrebarea :
M.T.:“Mai sunt dispusi tinerii de astazi sa invete matematica?”
I.T.:”Daca vom sti sa prezentam aceasta stiinta ca pe o frumoasa provocare a
spiritului mereu nascocitor, este posibil ca tinerii sa ajunga sa inteleaga
frumusetea si profunzimea unui rationament matematic.Si mai trebuie ca
profesorii sa fie capabili sa prezinte elevilor impactul matematicii asupra intregii
dezvoltari stiintifice contemporane, conexiunile dintre matematica si
informatica si aplicatiile acestora, de exemplu, in criptografie, in studiul
genomului uman, in comertul electronic.”
Bibliografie
[ 1 ]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaaIXaaacaGLBbGaayzxaaaaaa@38A0@
∗∗∗
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4fIOIaey4fIOIaey4fIOcaaa@38C0@
Gazeta Matematica, 1895
–
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@
2007 .
[ 2 ]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaaIYaaacaGLBbGaayzxaaaaaa@38A1@
Fauvel, J., & van Maanen, J., History in Mathematics Education,
Boston, 2000.
[ 3 ]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaaIZaaacaGLBbGaayzxaaaaaa@38A2@
Finch,S.R., Mathematical Constants, Cambridge University, 2003.
[ 4 ]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaaI0aaacaGLBbGaayzxaaaaaa@38A3@
Knot, R., Fibonacci
Numbers and the Golden Section, se poate consulta gratuit pe Internet la
adresa http://www.mcs.surrey.ac.uk/Personal/R.Knot/Fibonacci/fib.html
[ 5 ]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaaI1aaacaGLBbGaayzxaaaaaa@38A4@
Mihaileanu, N., Istoria Matematicii, vol. 1, Editura Enciclopedica Romana, Bucuresti,
1974.
[ 6 ]
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaaI2aaacaGLBbGaayzxaaaaaa@38A5@
Mihaileanu, N., Istoria Matematicii, vol. 2, Editura Stiintifica si Enciclopedica,
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MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaaI3aaacaGLBbGaayzxaaaaaa@38A6@
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MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacaaI4aaacaGLBbGaayzxaaaaaa@38A7@
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MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@
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MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C0@
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