An Extension of a Congruence by Tauraso. (25th December 2012) - Meštrović, Romeo
Abstract : For a positive integer n let H n = ∑ k = 1 n 1 / n be the n th harmonic number. In this paper we prove that, for any prime p ≥ 7, ∑ k = 1 p - 1 H k / k 2 ≡ ∑ k = 1 p - 1 H k 2 / k ≡( 3 / 2 p) ∑ k = 1 p - 1 1 / k 2 ( mod p 2 ) . Notice that the first part of this congruence is proposed in 2008 by Tauraso. In our elementary proof of the second part of the above congruence we use certain classical congruences modulo a prime and the square of a prime, some congruences involving harmonic numbers, and a combinatorial identity due to Hernández. Our auxiliary results contain many interesting combinatorial congruences involving harmonic numbers.