# Fibonacci numbers and Fermat's last theorem

Acta Arithmetica (1992)

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

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## Abstract

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Let Fₙ be the Fibonacci sequence defined by F₀=0, F₁=1, ${F}_{n+1}=Fₙ+{F}_{n-1}\left(n\ge 1\right)$. It is well known that ${F}_{p-\left(5/p\right)}\equiv 0\left(modp\right)$ for any odd prime p, where (-) denotes the Legendre symbol. In 1960 D. D. Wall  asked whether $p²|{F}_{p-\left(5/p\right)}$ is always impossible; up to now this is still open. In this paper the sum ${\sum }_{k\equiv r\left(mod10\right)}\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ is expressed in terms of Fibonacci numbers. As applications we obtain a new formula for the Fibonacci quotient ${F}_{p-\left(5/p\right)}/p$ and a criterion for the relation $p|{F}_{\left(p-1\right)/4}$ (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.

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