Keywords:
1. Introduction
At the beginning of the 21th century, Zhao (Cf. [1] ) first announced the following curious congruence involving multiple harmonic sums for any odd prime
,
(1)
which holds when
evidently. Here, Bernoulli numbers
are defined by the recursive relation:
A simple proof of (1) was presented in [2]. This congruence has been generalized along several directions. First, Zhou and Cai [3] established the following harmonic congruence for prime
and integer
(2)
Later, Xia and Cai [4] generalized (1) to
where
is a prime.
Recently, Wang and Cai [5] proved for every prime
and positive integer r,
(3)
where
denotes the set of positive integers which are prime to p.
Let
or 4, for every positive integer
and prime
, Zhao [6] extended (3) to
(4)
For any prime
and integer
, Wang [7] proved that
We consider the following alternating harmonic sums
where
. Given n, we only need to consider the following alternating harmonic sums,
where
denotes the largest integer less than or equal to x.
In this paper, we consider the congruences involving the combination of alternating harmonic sums,
We obtain the following theorems. Among them, Theorem 1 and Theorem 2 have been proved by Wang [8] using different method.
Theorem 1. Let p be an odd prime and r a positive integer, then
Remark 1. There is no solution
for the equation
with
.
Theorem 2. Let p be an odd prime and r a positive integer, then
Theorem 3. Let
be a prime and r a positive integer, then
Theorem 4. Let
be a prime and r a positive integer, then
2. Preliminaries
In order to prove the theorems, we need the following lemmas.
Lemma 1 ( [5] ) Let p be an odd prime and r, m positive integers, then
Lemma 2. Let p be an odd prime and r, m positive integers, then
Proof. It is easy to see that
Let
, then
and
. By symmetry, we have
This completes the proof of Lemma 2. ¨
Lemma 3. Let
be a prime and r, m positive integers, then
Proof. The proof of Lemma 3 is similar to the proof of Lemma 2. ¨
Lemma 4 ( [3] ) Let
be positive integers,
, then
Lemma 5 ( [7] ). Let p be an odd prime, and
positive integers, where
, then
Lemma 6. Let
be a prime, then
Proof. By Lemma 3, we have
(5)
It is easy to see that
By Lemma 4, we have
(6)
Hence
Replace
, then
and
Thus
(7)
Using Lemma 5 in the first sum of the right hand in (7) and using Lemma 4 in the second sum, we have
(8)
Combining (5) with (8), we complete the proof of Lemma 6. ¨
Lemma 7. Let
be a prime and
a positive integer, then
Proof. The proof of Lemma 7 is similar to the proof method of (4) in [6]. ¨
Lemma 8 ( [7] ). Let
be a prime and
positive integers,
, then
Lemma 9. Let
be a prime and
positive integers, then
Proof. The proof of Lemma 9 is similar to the proof of Lemma 2. ¨
3. Proofs of the Theorems
Proof of Theorem 1. It is easy to see that
(9)
Let
,then
and
, hence
(10)
Let
, then
and
, hence
(11)
Noting that
,
and we rename
to
, then
(12)
Rename i to j and
to
, then
(13)
Combining (9)-(13), we have
(14)
Let
in the first sum of (14) and noting that
(14) is equal to
(15)
By Lemma 1, Lemma 2 and (15), we obtain
This completes the proof of Theorem 1. ¨
Proof of Theorem 2. For every triple
of positive integers which satisfies
, we take it to 3 cases.
Cases 1. Let
.
is a bijection between the solutions of
and
, we have
(16)
Cases 2. Let
.
is a bijection between the solutions of
and
, we have
(17)
Cases 3. Let
and
in the former and
in the later are the bijections between the solutions of
and
, we have
(18)
Combining (16)-(18), we have
By Theorem 1, we complete the proof of Theorem 2. ¨
Proof of Theorem 3. By symmetry, it is easy to see that
(19)
Let
in the first sum of the last equation in (19), then
, (19) equals to
(20)
Let
in the second sum of the last equation in (20), since
, then
, (20) equals to
Similarly, we have
Hence
By Lemma 3, we have
By (2) and Lemma 6, we have
By (4) and Lemma 7, if
, then
This completes the proof of Theorem 3. ¨
Proof of Theorem 4. Similar to the proofs of Theorem 1 and Theorem 3, we have
and
Hence
By Lemma 9, we have
By (2) and Lemma 8 (1), we have
By Lemma 8 (2), if
, then
This completes the proof of Theorem 4. ¨
4. Conclusions
Let p be an odd prime and
positive integers,
, using Lemma 1 and Lemma 2, similar to the proof of Theorem 1, we can prove that
In particular, if
, it becomes Theorem 1.
Let p be odd prime and
positive integers,
, similar to the proof of Theorem 2, we can prove that
In particular, if
, it becomes Theorem 2.
Let
be a prime and
positive integers,
, we can deduce the congruence
for
Let
be a prime and
positive integers,
, we can deduce the congruence
for
Similarly, we can consider the congruence
for
where
, but it seems much more complicated.
Founding
This work is supported by the Natural Science Foundation of Zhejiang Province, Project (No. LY18A010016) and the National Natural Science Foundation of China, Project (No. 12071421).