On a Supercongruence Conjecture of Z.-W.Sun∗

Guo-shuai MAO

1Department of Mathematics,Nanjing University of Information Science and Technology,Nanjing 210044,China.E-mail: maogsmath@163.com

Abstract In this paper,the author partly proves a supercongruence conjectured by Z.-W.Sun in 2013.Let p be an odd prime and let a ∈Z+.Then,if p≡1 (mod 3),

is obtained,where()is the Jacobi symbol.

Keywords Supercongruences,Binomial coefficients,Fermat quotient,Jacobi symbol

1 Introduction

In the past years,congruences for sums of binomial coefficients have attracted the attention of many researchers (see,for instance,[1,3–4,6,10,12,16–17,19]).In 2011,Sun [17]proved that for any odd primepanda∈Z+,

Recently,Liu and Petrov [7]showed some congruences on sums ofq-binomial coefficients.

Pan and Sun [13]proved that for any primep≡1 (mod 4) or 1

In 2017,Mao and Sun [11]showed that for any primep≡1 (mod 4) or 1

Sun [15]proved that for any odd primepanda∈Z+,

In this paper,we partly prove Sun’s conjecture (see [15,Conjecture 1.2(i)]).

Theorem 1.1Let p be an odd prime and let a∈Z+.If p≡1 (mod 3),then

We shall prove Theorem 1.1 in Section 2.Our result is much interesting because of much rarer are the examples where the upper limit of the sum is strictly betweenandp−1,and these congruences are much more difficult to handle.

2 Proof of Theorem 1.1

Lemma 2.1(see [5])For any prime p>3,we have the following congruences modulo p

Proof of Theorem 1.1In view of (1.1),we just need to verify that

Letkandlbe positive integers withk+l=paand 0

and

So we have

It is easy to see that fork=1,2,···,

This,with Fermat’s little theorem yields that

Thus,by (2.1) we only need to show that

Now we set

then we only need to prove that

Settingn=n−1 in the last equation of page 3 in [18],we have

It is easy to check that for each 0≤k≤n−1−m,

whereB(P,Q) stands for the beta function.It is well known that the beta function relates to gamma function:

So

Therefore

Hence,by (2.6),we just need to show that

It is obvious that

where

By the following transformation

we have

In view of [2,(1.48)],we have

and it is easy to check that

Thus

Now we calculate C.First we have the following transformation

Hence,

With the help of package Sigma (see [14]),we find the following identity:

which can be easily proved by induction onN.

SubstitutingN=n−m−1,i=m+1−jinto the above identity,we have

It is easy to check that

Therefore

Hence

That is

One can easily check that

and (the following identity can be found in [2])

These yield that

Replacingnbymin the above equation,we have

Hence

So

In view of [16,(1.20)],and by (2.2)–(2.4) we have

It is obvious that

Sincep≡1 (mod 3),by [8,Lemma 17(2)],we have

These,with (2.7)–(2.10) yield that we only need to prove the following congruence:

There are only the items 3j+1=pa−1(3k+1) withk=0,1,···,so,by [9,Theorem 1.2]and Lucas congruence,we have

Therefore the proof of Theorem 1.1 is complete.

AcknowledgementThe author would like to thank the anonymous referees for helpful comments.