Geodesic
Fibonacci numbers and Fermat's last theorem
Acta Arithmetica, Tome 60 (1991) no. 4, pp. 371-388.

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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 $∑_{k≡ r (mod 10)}{n\choose k}$ 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.
DOI : 10.4064/aa-60-4-371-388

Zhi-Wei Sun 1

1 
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Zhi-Wei Sun. Fibonacci numbers and Fermat's last theorem. Acta Arithmetica, Tome 60 (1991) no. 4, pp. 371-388. doi : 10.4064/aa-60-4-371-388. http://geodesic.mathdoc.fr/articles/10.4064/aa-60-4-371-388/

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