Proof of three conjectures on congruences

In this paper we prove three conjectures on congruences involving central binomial coefficients or Lucas sequences. Let $p$ be an odd prime and let $a$ be a positive integer. We show that if $p\equiv 1\pmod{4}$ or $a>1$ then $\sum_{k=0}^{\lfloor\frac34p^a\rfloor}\binom{-1/2}k\equiv\left(\frac{2}{p^a}\right)\pmod{p^2},$ where $(-)$ denotes the Jacobi symbol. This confirms a conjecture of the second author. We also confirm a conjecture of R. Tauraso by showing that $\sum_{k=1}^{p-1}\frac{L_k}{k^2}\equiv0\pmod{p}\quad {\rm provided}\ \ p>5,$ where the Lucas numbers $L_0,L_1,L_2,\ldots$ are defined by $L_0=2,\ L_1=1$ and $L_{n+1}=L_n+L_{n-1}\ (n=1,2,3,\ldots)$. Our third theorem states that if $p\not=5$ then we can determine $F_{p^a-(\frac{p^a}5)}$ mod $p^3$ in the following way: $\sum_{k=0}^{p^a-1}(-1)^k\binom{2k}k\equiv\left(\frac{p^a}5\right)\left(1-2F_{p^a-(\frac{p^a}5)}\right)\ \pmod{p^3},$ which appeared as a conjecture in a paper of Sun and Tauraso in 2010.

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