Fibonacci numbers and Fermat's last theorem

Zhi-Wei Sun

Acta Arithmetica (1992)

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

Abstract

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Let Fₙ be the Fibonacci sequence defined by F₀=0, F₁=1, F n + 1 = F ₙ + F n - 1 ( n ≥ 1 ) . It is well known that F p - ( 5 / p ) ≡ 0 ( m o d p ) for any odd prime p, where (-) denotes the Legendre symbol. In 1960 D. D. Wall [13] asked whether p ² | F p - ( 5 / p ) is always impossible; up to now this is still open. In this paper the sum ∑ k ≡ r ( m o d 10 ) n 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.

How to cite

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Zhi-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

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  1. [1] L. E. Dickson, History of the Theory of Numbers, Vol. I, Chelsea, New York 1952, 105, 393-396. 
  2. [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. [3] E. Lehmer, On the quartic character of quadratic units, J. Reine Angew. Math. 268/269 (1974), 294-301. Zbl0289.12007
  4. [4] L. J. Mordell, Diophantine Equations, Academic Press, London and New York 1969, 60-61. 
  5. [5] P. Ribenboim, 13 Lectures on Fermat's Last Theorem, Springer, New York 1979, 139-159. 
  6. [6] Zhi-Hong Sun, Combinatorial sum ∑ k = 0 k ≡ r ( m o d m ) n n k and its applications in number theory (I), J. Nanjing Univ. Biquarterly, in press. 
  7. [7] Zhi-Hong Sun, Combinatorial sum ∑ k = 0 k ≡ r ( m o d m ) n n k and its applications in number theory (II), J. Nanjing Univ. Biquarterly, in press. 
  8. [8] Zhi-Wei Sun, A congruence for primes, preprint, 1991. 
  9. [9] Zhi-Wei Sun, On the combinatorial sum ∑ k ≡ r ( m o d m ) n k , submitted. 
  10. [10] Zhi-Wei Sun, Combinatorial sum ∑ k ≡ r ( m o d 12 ) n k and its number-theoretical applications, to appear. 
  11. [11] Zhi-Wei Sun, Reduction of unknowns in Diophantine representations, Science in China (Ser. A) 35 (1992), 1-13. 
  12. [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. [13] D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly 67 (1960), 525-532. Zbl0101.03201
  14. [14] H. C. Williams, A note on the Fibonacci quotient F p - ε / p , Canad. Math. Bull. 25 (1982), 366-370 Zbl0491.10009

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