Abstract

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].

2. Proof of Theorem 1

The following congruences given by Sun in his recent paper [6] are needed in the proof of Theorem 1.

Lemma 4. Let be a prime. Then for every .

Proof. The congruences (9) and (10) are in fact the congruences (2.1) and (2.2) in [6, Lemma 2.1], respectively.

The following well-known result is a generalization of Wolstenholme's theorem (see, e.g., [10, Theorem 1] or [11]).

Lemma 5 (see [12, Theorem 3]). Let be a positive integer, and let be a prime such that . Then In particular, for any prime , and for any prime , and .

Lemma 6. Let be a prime. Then

Proof. By the congruence (9) from Lemma 4, for each (notice that this is true for because ), and therefore Furthermore, using (11) with we get From (17) and (18) it follows that which is actually (13).
Since , for each , we have The above identity and (13) and (11) of Lemma 5 with yield On the other hand, since by (9) from Lemma 4, for each , then Taking (22) into (21) gives which proves (14).
Proof of the congruence (15) is completely analogous to the previous proof using the fact that, by Lemma 5, and therefore, for each Finally (cf. [2]), from the identity immediately follows that Inserting in the right hand side of the identity (26) the congruences given in Lemma 5, we immediately obtain (16). This completes the proof.

Lemma 7. Let be a prime. Then

Proof. Since , for every , we get Using particular congruences given in Lemma 5 with and , we find that Substituting the congruences (29), (13) of Lemma 6, and (11) with of Lemma 5 into (28), we obtain The right hand side of (30) can be expressed as Taking (15) of Lemma 6 into (31) and comparing this with (30), we immediately obtain (27).

Further, for the proof of Theorem 1 we will need two particular cases of the following identity due to Hernández [4].

Lemma 8 (see [4]). Let and be positive integers. Then

Lemma 9. Let be a prime. Then

Proof. The identity (32) of Lemma 8 with and becomes For any fixed , we have the identity Next the congruence (10) from Lemma 4 reduced modulo gives for every . Substituting (35), (36), and the congruence of Lemma 5 into (34), we immediately obtain or equivalently, Further, (16) from Lemma 6 and the congruences from Lemma 5 give Substituting (39) into (38), we find that Taking (14) of Lemma 6 into (40) yields (33). This concludes the proof.

Lemma 10. Let be a prime. Then

Proof. For simplicity, we denote Obviously, the following identity holds The well-known Newton's identities (see, e.g., [13]) imply whence since all the sums , and are divisible by a prime , we obtain (cf. [5, Theorem 1.5] or [14]) Inserting (45) and into (43), we get Further, by the substitution trick , From (47) we see that , which substituting into (46) gives Finally, (47) and (48) yield , as desired.

Lemma 11. Let be a prime. Then

Proof. We follow proof of the congruence (3) in Theorem 1.1 of [15]. By Lemma 5, , or equivalently, for each holds Applying the congruence (50), we find that whence it follows that Since by the first part of (13) from Lemma 6 and by (11) of Lemma 5 with , substituting this into (52), we obtain (49).