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 ≡ 1 (mod 4) or a > 1 then $\sum\limits_{k = 0}^{\left\lfloor {\tfrac{3} {4}p^a } \right\rfloor } {\left( {\tfrac{{ - 1/2}} {k}} \right) \equiv \left( {\tfrac{2} {{p^a }}} \right)(\bmod 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 $\sum\limits_{k = 1}^{\left\lfloor {p - 1} \right\rfloor } {\tfrac{{L_k }} {{k^2 }} \equiv 0(\bmod p)providedp > 5,} $ where the Lucas numbers L 0, L 1, L 2, … are defined by L 0 = 2, L 1 = 1 and L n+1 = L n +L n−1 (n = 1, 2, 3, …). The third theorem states that if p ≠ 5 then $F_{p^a - (\tfrac{{p^a }} {5})} $ mod p 3 can be determined in the following way: $\sum\limits_{k = 0}^{p^a - 1} {( - 1)^k (_k^{2k} ) \equiv \left( {\tfrac{{p^a }} {5}} \right)(1 - 2F_{p^a - (\tfrac{{p^a }} {5})} )(\bmod p^3 ),} $ which appeared as a conjecture in a paper of Sun and Tauraso in 2010.