FIBONACCI NUMBERS AND FERMAT LAST THEOREM

Let {F(n)} be the Fibonacci sequence defined by F0 = 0, F1 = 1, F(n) + 1 = F(n) + F(n-1) (n greater-than-or-equal-to 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 [GRAPHICS] 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 not-equal 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.