An Extension of a Congruence by Tauraso

For a positive integer let be the th harmonic number. In this paper we prove that, for any prime ,？？ . 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. 1. Introduction and Main Results Given positive integers and , the harmonic numbers of order are those rational numbers defined as For simplicity, we will denote by the th harmonic number (we assume in addition that ). Usually, here as always in the sequel, we consider the congruence relation modulo a prime extended to the ring of rational numbers with denominators not divisible by . For such fractions we put if and only if , and the residue class of is the residue class of where is the inverse of modulo . By a problem proposed by Tauraso in [1] and recently solved by Tyler [2], for any prime , Further, Tauraso [3, Theorem 2.3] proved Tauraso's proof of (4) is based on an identity due to Hernández [4] (see Lemma 8) and the congruence for triple harmonic sum modulo a prime due to Zhao [5] (see (64) of Remarks in Section 2). In this paper, we give an elementary proof of (4) and its extension as follows. Theorem 1. If is a prime, then Recall that Sun in [6] established basic congruences modulo a prime for several sums of terms involving harmonic numbers. In particular, Sun established for . Further generalizations of these congruences are recently obtained by Tauraso in [7]. Recall that the Bernoulli numbers are defined by the generating function It is easy to find the values , , , , and for odd . Furthermore, for all . Applying a congruence given in [8, Theorem ] related to the sum modulo , the congruence (5) in terms of Bernoulli numbers may be written as follows. Corollary 2. Let be a prime. Then In particular, one has Remark 3. Notice that the second congruence of (8) was obtained by Sun and Tauraso [9, the congruence (5.4)] by using a standard technique expressing sum of powers in terms of Bernoulli numbers. Our proof of the second part of the congruence (5) given in the next section is entirely elementary and it is combinatorial in spirit. It is based on certain classical congruences modulo a prime and the square of a prime, two simple congruences given by Sun [6], and two particular cases of a combinatorial identity due to Hernández [4].