On congruences related to central binomial coefficients

By Zhi-Wei Sun. 2009

115 citations recorded by the source corpus.

It is known that $\sum_{k=0}^\infty\binom{2k}{k}/((2k+1)4^k)=\pi/2$ and $\sum_{k=0}^\infty\binom{2k}{k}/((2k+1)16^k)=\pi/3$. In this paper we obtain their p-adic analogues such as $$\sum_{p/2 3 is a prime and E_0,E_1,E_2,... are Euler numbers. Besides these, we also deduce some other congruences related to central binomial coefficients. In addition, we pose some conjectures one of which states that for any odd prime p we have $$\sum_{k=0}^{p-1}\binom{2k}{k}^3=4x^2-2p (mod p^2)$$ if (p/7)=1 and p=x^2+7y^2 with x,y integers, and $$\sum_{k=0}^{p-1}\binom{2k}{k}^3=0 (mod p^2)$$ if (p/7)=-1, i.e., p=3,5,6 (mod 7).

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/3b01262f5f0d5ca546ba1f8b8b83b15218ded88e

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

Citations in the corpus

Some congruences related to a congruence of Van Hamme (2019, background): “[26] Sun Z-W.”
Binomial coefficients, Catalan numbers and Lucas quotients (2009, background): “also [SSZ], [GZ] and [ S09b ] for related results on p-adic valuations.”