Abstract
Define the sums of triple reciprocals . Zhao discovered the following curious congruence for any odd prime , Xia and Cai extended the above congruence to modulo where is a prime. In this paper, we consider the congruences about (where is the integral part of , ) modulo . When , the results we obtain are the results of Zhao and Xia and Cai.
1. Introduction
Letwhere denotes the set of positive integers which are prime to .
In 2003, Zhao [1] first announced (see [2], Corollary 4.2) the following curious congruence involving multiple harmonic sums for any odd prime :which holds when evidently. Here, Bernoulli numbers are defined by the recursive relation:
The first few Bernoulli numbers are ,
A simple proof of (2) was presented in [3]. This congruence has been generalized along several directions. First, Zhou and Cai [4] established the following harmonic congruence for prime and integer :
Later, Xia and Cai [5] generalized (2) towhere is a prime.
In 2014, Wang and Cai [6] proved for every prime and positive integer ,
Let or 4; for every positive integer and prime , Zhao [7] generalized (7) to
In 2017, Cai et al. [8] replaced the odd prime in the summation and modulus with a product of two odd prime powers. For other related papers, see [9–11].
In 2022, Chern [12] established the generalization of the conjecture in [8].
We have seen that the sum of the subscript variables in the literature is an integer multiple of the modulus. Next, we are going to consider the case where the sum of the subscript variables is not an integer multiple of the modulus. When positive integer , we have
In this paper, we consider the congruences about (where is the integral part of , ) modulo and obtain the following theorems.
Let be defined bywhere is the -th harmonic number .
Theorem 1. Let be a positive integer greater than 2. Then,
Theorem 2. Let be a prime greater than 5. Then,andwhere is the Fermat quotient. More generally, we havewhere are the Bernoulli polynomials defined byand is the Legendre symbol and is the fractional part of .
From Theorem 2, we obtainand this is the result of (2), and
From Theorem 2, by Lemmas 6 and 7 (see Auxiliary Results), we also obtain
Theorem 3. Let be a prime greater than 7. Then,where are Euler numbers defined by
From Theorem 3, we obtain
Theorem 4. Let be a prime greater than 11. Then,
From Theorem 4, by Lemmas 6 and 7 (see Auxiliary Results), we obtain
2. Auxiliary Results
Lemma 5 (see [13, 14]). Let be a prime and be a positive integer. Then,
Lemma 6 (see [15]). Let be a positive integer; then,
Lemma 7 (see [15]). Let be an odd prime, , and be positive integers with . Then,
Lemma 8 (see [16]). Let be a prime greater than 5. Then,
Lemma 9 (see [16]). Let be a prime greater than 5. Then,
Lemma 10 (see [16]). Let be a prime greater than 5. Then,
3. Proofs
3.1. Proof of Theorem 1
Proof. By symmetry, we haveLet in (30), and we getThus, we obtain Theorem 1.
Next, we consider the congruences about modulo .
3.2. Proof of Theorem 2
Proof. For , by Theorem 1 and Lemma 5, we haveThis is the result of (6). For , by Theorem 1 and Lemma 5, we haveFor , , by Theorem 1 and Lemma 8, we haveSince (see [16]) and by Lemma 6 in (34), we haveThus, we obtain Theorem 2.
3.3. Proof of Theorem 3
Proof. For , , by Theorem 1 and Lemma 9, we haveThus, we obtain Theorem 3.
3.4. Proof of Theorem 4
Proof. For , , by Theorem 1 and Lemma 10, we haveThus, we obtain Theorem 4.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
This study was supported by the National Natural Science Foundation of China (Project No. 12071421).