FIBONACCI NUMBERS MODULO CUBES OF PRIMES - Sun, Zhi-Wei (Taiwanese journal of mathematics, 2013)
Letpbe an odd prime. It is well known that F p − ( p 5 ) ≡ 0 (modp), where Fn n⩾0is the Fibonacci sequence and (–) is the Jacobi symbol. In this paper we show that ifp≠ 5 then we may determine F p − ( p 5 ) modp3 in the following way: ∑ k = 0 ( p − 1 ) / 2 ( k 2 k ) ( − 16 ) k ≡ ( p 5 ) ( 1 + F p − ( p 5 ) 2 ) ( mod p 3 ) . We also use Lucas quotients to determine ∑ k = 0 ( p − 1 ) / 2 ( k 2 k ) / m k modulop 2for any integerm≢ 0 (modp); in particular, we obtain ∑ k = 0 ( p − 1 ) / 2 ( k 2 k ) 16 k ≡ ( 3 p ) ( mod p 2 ) . In addition, we pose three conjectures for further research. 2010Mathematics Subject Classification: Primary 11B39, 11B65; Secondary 05A10, 11A07. Key words and phrases: Fibonacci numbers, Central binomial coefficients, Congruences, Lucas sequences.