On congruences related to central binomial coefficients

Abstract

It is known that ${\sum }_{k=0}^{\infty }\left(\genfrac{}{}{0ex}{}{2k}{k}\right)/((2k+1)4^k)=\pi /2$ and ${\sum }_{k=0}^{\infty }\left(\genfrac{}{}{0ex}{}{2k}{k}\right)/((2k+1)16^k)=\pi /3$. In this paper we obtain their p-adic analogues such as $$\sum _{p/2<k<p}\left(\genfrac{}{}{0ex}{}{2k}{k}\right)/((2k+1)4^k)=3\sum _{p/2<k<p}\left(\genfrac{}{}{0ex}{}{2k}{k}\right)/((2k+1)16^k)=p{E}_{p-3}(mod{p}^2),$$ where p>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 $$\sum _{k=0}^{p-1}{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^3=4x^2-2p(mod{p}^2)$$ if (p/7)=1 and p=x^2+7y^2 with x,y integers, and $$\sum _{k=0}^{p-1}{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^3=0(mod{p}^2)$$ if (p/7)=-1, i.e., p=3,5,6 (mod 7).

References in corpus (6)

• On Delannoy numbers and Schröder numbers
• Various congruences involving binomial coefficients and higher-order Catalan numbers
• On sums of binomial coefficients modulo p^2
• p-adic valuations of some sums of multinomial coefficients
• Fibonacci numbers modulo cubes of primes
• Binomial identities related to Calabi-Yau differential equations

Cited by in corpus (12)

• Open Conjectures on Congruences
• New congruences involving products of two binomial coefficients
• New series for some special values of $L$-functions
• On sums involving products of three binomial coefficients
• Conjectures and results on $x^2$ mod $p^2$ with $4p=x^2+d{y}^2$
• On sums of Apéry polynomials and related congruences
• Fibonacci numbers modulo cubes of primes
• A new series for ${\pi }^3$ and related congruences
• An Extension of a Congruence by Kohnen
• Congruences involving binomial coefficients and Lucas sequences
• Curious congruences for Fibonacci numbers
• A search for primes $p$ such that Euler number $E_{p-3}$ is divisible by $p$