A pair of congruences concerning sums of central binomial coefficients

Guo-Shuai Mao, Roberto Tauraso

Published 2020-04-20Version 1

In this paper, we prove the following congruences: for any prime $p\equiv1\pmod3$: $$\sum_{k=1}^{\lfloor\frac{2p}3\rfloor}\binom{2k}{k}(-2)^k\equiv0\pmod{p^2}\qquad \text{ and }\qquad \sum_{k=0}^{\lfloor\frac{5p}6\rfloor}\frac{\binom{2k}k}{(-32)^k}\equiv\left(\frac2p\right)\pmod{p^2}$$ where $\left(\frac{\cdot}{p}\right)$ stands for the Legendre symbol.