FIBONACCI NUMBERS MODULO CUBES OF PRIMES

Let p be an odd prime. It is well known that Fp-(p/5) 0 (mod p), where {F-n}(n >= 0) is the Fibonacci sequence and (-) is the Jacobi symbol. In this paper we show that if p not equal 5 then we may determine Fp-(p/5) mod p(3) in the following way: Sigma((p-1)/2)(k=0) (2k k)/(-16)(k) (p/5) (1 + Fp-(p/5)/2) (mod p(3)). We also use Lucas quotients to determine Sigma((p-1)/2)(k=0) (2k k)/m(k) modulo p(2) for any integer m (sic) 0 (mod p); in particular, we obtain Sigma((p-1)/2)(k=0) (2k k)/16(k) = (3/p) (mod p(2)). In addition, we pose three conjectures for further research.