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Fibonacci numbers and Fermat's last theorem

Zhi-Wei Sun (1992)

Acta Arithmetica

Let Fₙ be the Fibonacci sequence defined by F₀=0, F₁=1, F n + 1 = F + 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...

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