Proof of a conjecture of Adamchuk

Abstract

In this paper, we prove a congruence which confirms a conjecture of Adamchuk. For any prime $p\equiv 1(\mathrm{mod}3)$ and $a\in {\mathbb{Z}}^{+}$, we have \begin{align*} \sum_{k=1}^{\frac{2}3(p^a-1)}\binom{2k}k\equiv0\pmod{p^2}. \end{align*}

14 pages

References in corpus (1)

• Congruences on sums of $q$-binomial coefficients

Cited by in corpus (2)

• On a supercongruence conjecture of Z.-W. Sun
• Three pairs of congruences concerning sums of central binomial coefficients