Five Curious Congruences Modulo p2

Abstract

Abstract Let p ≥ 3 be a prime, and let qp(2) := (2P 1 - 1)/p be the Fermat quotient of p to base 2. In this note, we prove that . As an application, using two combinatorial identities due to T. B. Staver in 1947 and G. Galperin and H. Gauchman in 2004, we obtain four curious combinatorial congruences modulo p2. As an auxiliary result, here we present an elementary proof of a congruence established by E. Lehmer in 1938. Notice that this con­gruence together with some other congruences given in our lemmas leads to the elementary proof of a beautiful Morley’s congruence due to F. Morley in 1895.