Fibonacci numbers and Fermat's last theorem

Zhi-Wei Sun

Zhi-Wei Sun

Acta Arithmetica, Tome 62 (1992), p. 371-388 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $F_{p - (5 / p)} \equiv 0 (m o d p)$ 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 $\sum_{k \equiv r (m o d 10)} (\binom{n}{k})$ 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.

Publié le : 1992-01-01

Zbl 0725.11009

EUDML-ID : urn:eudml:doc:206445

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     title = {Fibonacci numbers and Fermat's last theorem},
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     year = {1992},
     pages = {371-388},
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Zhi-Wei Sun. Fibonacci numbers and Fermat's last theorem. Acta Arithmetica, Tome 62 (1992) pp. 371-388. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav60i4p371bwm/

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