Proof of three conjectures on congruences (1010.2489v5)
Abstract: 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 $$ \sum_{k=0}{\lfloor\frac34pa\rfloor}\binom{-1/2}k\equiv\left(\frac{2}{pa}\right)\pmod{p2}, $$ where $(-)$ denotes the Jacobi symbol. This confirms a conjecture of the second author. We also confirm a conjecture of R. Tauraso by showing that $$\sum_{k=1}{p-1}\frac{L_k}{k2}\equiv0\pmod{p}\quad {\rm provided}\ \ p>5,$$ 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 $F_{pa-(\frac{pa}5)}$ mod $p3$ in the following way: $$\sum_{k=0}{pa-1}(-1)k\binom{2k}k\equiv\left(\frac{pa}5\right)\left(1-2F_{pa-(\frac{pa}5)}\right)\ \pmod{p3},$$ which appeared as a conjecture in a paper of Sun and Tauraso in 2010.