ISRN Combinatorics

Volume 2013 (2013), Article ID 363724, 7 pages

http://dx.doi.org/10.1155/2013/363724

Research Article

An Extension of a Congruence by Tauraso

Romeo Meštrović

Department of Mathematics, Maritime Faculty, University of Montenegro, Do-Brota 36, 85330 Kotor, Montenegro

Received 23 September 2012; Accepted 10 October 2012

Academic Editors: C. da Fonseca and B. Wu

Copyright © 2013 Romeo Meštrović. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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).

The first congruence of the following result was recently established by Sun [6, Theorem 1.1, congruence (1.5)].

Lemma 12. Let be a prime. Then

Proof. Comparing the congruences (27) of Lemma 7 and (33) of Lemma 9, we have

Since by (41) of Lemma 10,

then substituting this into (57), we obtain

Finally, as by (49) of Lemma 11,

then comparing this with (59) implies

which coincides with (54).

Finally, (54) and (33) of Lemma 9 yield (55), while (54) and (14) of Lemma 6 yield (56).

Remarks 13. Applying a standard technique expressing sum of powers in terms of Bernoulli numbers, Sun in [6, proof of (1.5) of Theorem 1.1] showed that

The above congruence and (54) yield the following curious congruence for a prime established by Zhao [5, congruence (3.19) of Corollary 3.6]:

As noticed in [6, proof of Lemma 2.8], the above congruence immediately follows from an identity of Matiyasevich (cf. [16, equation (1.3)]).

Furthermore, the congruences (54), (59), and (41) from Lemma 10 immediately give

Notice that the congruences (64) were proved by Zhao [5, Corollary 3.6, congruence (3.20)] applying a technique expressing sum of powers in terms of Bernoulli numbers.

The following result is contained in [17, Lemma 2.4].

Lemma 14. Let be a prime. Then

Proof. Multiplying the identity

by (), we obtain

which can be written as

After summation of the above congruence over we immediately obtain

Taking the congruences and of Lemma 5 into (69), it becomes (65).

Proof of Theorem 1. The first congruence of (5) is in fact the congruence (16) of Lemma 6.

The identity (32) of Lemma 8 with and becomes

Substituting the congruence (10) of Lemma 4 into the left hand side of equality (70), we find that

By Wolstenholme's theorem,

Next we have

Further, using (56) of Lemma 12,

Similarly, using (55) of Lemma 12,

Now inserting (72)–(75) into (71), we obtain

The equality (70) and the congruence (76) give

The congruences (77) and (65) of Lemma 14 yield

Finally, the congruences (77) and (78) complete proof of Theorem 1.

Remark 15. From the identity

immediately follows that

Inserting in the right hand side of the above identity the congruences and from [8, Theorem with ], we find that for a prime ,

However, the determination of seems to be a difficult problem.

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