Elsevier

Journal of Number Theory

Volume 131, Issue 11, November 2011, Pages 2219-2238
Journal of Number Theory

On congruences related to central binomial coefficients

https://doi.org/10.1016/j.jnt.2011.04.004Get rights and content
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Abstract

It is known thatk=0(2kk)(2k+1)4k=π2andk=0(2kk)(2k+1)16k=π3. In this paper we obtain their p-adic analogues such asp/2<k<p(2kk)(2k+1)4k3p/2<k<p(2kk)(2k+1)16kpEp3(modp2), where p>3 is a prime and E0,E1,E2, are Euler numbers. Besides these, we also deduce some other congruences related to central binomial coefficients. In addition, we pose some conjectures one of which states that for any odd prime p we havek=0p1(2kk)3{4x22p(modp2)if(p7)=1&p=x2+7y2(x,yZ),0(modp2)if(p7)=1,i.e.,p3,5,6(mod7).

MSC

primary
11B65
secondary
05A10
11A07
11B68
11E25

Keywords

Central binomial coefficients
Congruences modulo prime powers
Euler numbers
Binary quadratic forms

Cited by (0)

Supported by the National Natural Science Foundation (grant 10871087) and the Overseas Cooperation Fund (grant 10928101) of China.