Proof of three conjectures on congruences

This paper proves three conjectures on congruences involving central binomial coefficients or Lucas sequences. Let p be an odd prime and let a be a positive integer. It is shown that if p equivalent to 1 ( mod 4) or a > 1 then Sigma([3/4pa])(k=0) (-1/2 k) equivalent to (2 p(a)) (mod p(2)), where (-) denotes the Jacobi symbol. This confirms a conjecture of the second author. A conjecture of Tauraso is also confirmed by showing that Sigma(p-1)(k=1) Lk/k(2) equivalent to 0 (mod p) provided p > 5, where the Lucas numbers L-0, L-1, L-2, ... are defined by L-0 = 2, L-1 = 1 and Ln+1 = L-n + Ln-1 (n = 1, 2, 3, ... ). The third theorem states that if p not equal 5 then F-pa - (p(a)/5) mod p(3) can be determined in the following way: Sigma(pa-1)(k=0) (-1)(k) (2k/k) (p(a)/5) (1 - 2F(pa - (pa/5))) (mod p(3)), which appeared as a conjecture in a paper of Sun and Tauraso in 2010