# Fibonacci numbers and Fermat's last theorem

Acta Arithmetica (1992)

- Volume: 60, Issue: 4, page 371-388
- ISSN: 0065-1036

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topZhi-Wei Sun. "Fibonacci numbers and Fermat's last theorem." Acta Arithmetica 60.4 (1992): 371-388. <http://eudml.org/doc/206445>.

@article{Zhi1992,

abstract = {Let Fₙ be the Fibonacci sequence defined by F₀=0, F₁=1, $F_\{n+1\}=Fₙ+F_\{n-1\} (n≥1)$. It is well known that $F_\{p-(5/p)\}≡ 0 (mod p)$ for any odd prime p, where (-) denotes the Legendre symbol. In 1960 D. D. Wall [13] asked whether $p²|F_\{p-(5/p)\}$ is always impossible; up to now this is still open.
In this paper the sum $∑_\{k≡ r (mod 10)\}\{n\atopwithdelims ()k\}$ is expressed in terms of Fibonacci numbers. As applications we obtain a new formula for the Fibonacci quotient $F_\{p-(5/p)\}/p$ and a criterion for the relation $p|F_\{(p-1)/4\}$ (if p ≡ 1 (mod 4), where p ≠ 5 is an odd prime. We also prove that the affirmative answer to Wall’s question implies the first case of FLT (Fermat’s last theorem); from this it follows that the first case of FLT holds for those exponents which are (odd) Fibonacci primes or Lucas primes.},

author = {Zhi-Wei Sun},

journal = {Acta Arithmetica},

keywords = {Fibonacci numbers; Legendre symbol; Lucas numbers; Fibonacci quotient; Fermat's last theorem; Fibonacci primes; Lucas primes},

language = {eng},

number = {4},

pages = {371-388},

title = {Fibonacci numbers and Fermat's last theorem},

url = {http://eudml.org/doc/206445},

volume = {60},

year = {1992},

}

TY - JOUR

AU - Zhi-Wei Sun

TI - Fibonacci numbers and Fermat's last theorem

JO - Acta Arithmetica

PY - 1992

VL - 60

IS - 4

SP - 371

EP - 388

AB - Let Fₙ be the Fibonacci sequence defined by F₀=0, F₁=1, $F_{n+1}=Fₙ+F_{n-1} (n≥1)$. It is well known that $F_{p-(5/p)}≡ 0 (mod p)$ for any odd prime p, where (-) denotes the Legendre symbol. In 1960 D. D. Wall [13] asked whether $p²|F_{p-(5/p)}$ is always impossible; up to now this is still open.
In this paper the sum $∑_{k≡ r (mod 10)}{n\atopwithdelims ()k}$ is expressed in terms of Fibonacci numbers. As applications we obtain a new formula for the Fibonacci quotient $F_{p-(5/p)}/p$ and a criterion for the relation $p|F_{(p-1)/4}$ (if p ≡ 1 (mod 4), where p ≠ 5 is an odd prime. We also prove that the affirmative answer to Wall’s question implies the first case of FLT (Fermat’s last theorem); from this it follows that the first case of FLT holds for those exponents which are (odd) Fibonacci primes or Lucas primes.

LA - eng

KW - Fibonacci numbers; Legendre symbol; Lucas numbers; Fibonacci quotient; Fermat's last theorem; Fibonacci primes; Lucas primes

UR - http://eudml.org/doc/206445

ER -

## References

top- [1] L. E. Dickson, History of the Theory of Numbers, Vol. I, Chelsea, New York 1952, 105, 393-396.
- [2] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, Oxford 1981, 148-150.
- [3] E. Lehmer, On the quartic character of quadratic units, J. Reine Angew. Math. 268/269 (1974), 294-301. Zbl0289.12007
- [4] L. J. Mordell, Diophantine Equations, Academic Press, London and New York 1969, 60-61.
- [5] P. Ribenboim, 13 Lectures on Fermat's Last Theorem, Springer, New York 1979, 139-159.
- [6] Zhi-Hong Sun, Combinatorial sum ${\sum}_{k=0k\equiv r\left(modm\right)}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ and its applications in number theory (I), J. Nanjing Univ. Biquarterly, in press.
- [7] Zhi-Hong Sun, Combinatorial sum ${\sum}_{k=0k\equiv r\left(modm\right)}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ and its applications in number theory (II), J. Nanjing Univ. Biquarterly, in press.
- [8] Zhi-Wei Sun, A congruence for primes, preprint, 1991.
- [9] Zhi-Wei Sun, On the combinatorial sum ${\sum}_{k\equiv r\left(modm\right)}\left(\genfrac{}{}{0pt}{}{n}{k}\right)$, submitted.
- [10] Zhi-Wei Sun, Combinatorial sum ${\sum}_{k\equiv r\left(mod12\right)}\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ and its number-theoretical applications, to appear.
- [11] Zhi-Wei Sun, Reduction of unknowns in Diophantine representations, Science in China (Ser. A) 35 (1992), 1-13.
- [12] H. S. Vandiver, Extension of the criteria of Wieferich and Mirimanoff in connection with Fermat's last theorem, J. Reine Angew. Math. 144 (1914), 314-318. Zbl45.0289.02
- [13] D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly 67 (1960), 525-532. Zbl0101.03201
- [14] H. C. Williams, A note on the Fibonacci quotient ${F}_{p-\epsilon}/p$ , Canad. Math. Bull. 25 (1982), 366-370 Zbl0491.10009

## Citations in EuDML Documents

top- Jiří Klaška, Short remark on Fibonacci-Wieferich primes
- Jiří Klaška, Criteria for testing Wall's question
- Mohamed Ayad, Périodicité (mod $q$) des suites elliptiques et points $S$-entiers sur les courbes elliptiques
- Jiří Klaška, Tribonacci modulo ${p}^{t}$
- Zhi-Hong Sun, On the theory of cubic residues and nonresidues
- Yann Bugeaud, Florian Luca, Maurice Mignotte, Samir Siksek, Almost powers in the Lucas sequence

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