Fibonacci numbers and Fermat's last theorem

Zhi-Wei Sun

ArticleOriginal scientific text

Title

Fibonacci numbers and Fermat's last theorem

Authors ¹

Affiliations

Department of Mathematics Nanjing University Nanjing 210008 People's Republic of China

Abstract

Let {Fₙ} be the Fibonacci sequence defined by F₀=0, F₁=1,

F_{n + 1} = F ₙ + F_{n - 1} (n \geq 1)

. It is well known that

F_{p - (5 / p)} \equiv 0 (mod 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

\sum_{k \equiv r (mod 10)} {n c h \infty s e k}

is expressed in terms of Fibonacci numbers. As applications we obtain a new formula for the Fibonacci quotient

\frac{F_{p - (\frac{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.

L. E. Dickson, History of the Theory of Numbers, Vol. I, Chelsea, New York 1952, 105, 393-396.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, Oxford 1981, 148-150.
E. Lehmer, On the quartic character of quadratic units, J. Reine Angew. Math. 268/269 (1974), 294-301.
L. J. Mordell, Diophantine Equations, Academic Press, London and New York 1969, 60-61.
P. Ribenboim, 13 Lectures on Fermat's Last Theorem, Springer, New York 1979, 139-159.
Zhi-Hong Sun, Combinatorial sum $\sum^{n}_{k = 0 k \equiv r (mod m)} {n c h \infty s e k}$ and its applications in number theory (I), J. Nanjing Univ. Biquarterly, in press.
Zhi-Hong Sun, Combinatorial sum $\sum^{n}_{k = 0 k \equiv r (mod m)} {n c h \infty s e k}$ and its applications in number theory (II), J. Nanjing Univ. Biquarterly, in press.
Zhi-Wei Sun, A congruence for primes, preprint, 1991.
Zhi-Wei Sun, On the combinatorial sum $\sum_{k \equiv r (mod m)} {n c h \infty s e k}$ , submitted.
Zhi-Wei Sun, Combinatorial sum $\sum_{k \equiv r (mod 12)} {n c h \infty s e k}$ and its number-theoretical applications, to appear.
Zhi-Wei Sun, Reduction of unknowns in Diophantine representations, Science in China (Ser. A) 35 (1992), 1-13.
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.
D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly 67 (1960), 525-532.
H. C. Williams, A note on the Fibonacci quotient $\frac{F_{p - ε}}{p}$ , Canad. Math. Bull. 25 (1982), 366-370

Title

Fibonacci numbers and Fermat's last theorem

Affiliations

Abstract

Bibliography