Some supercongruences involving

ABSTRACT

By applying a hypergeometric transformation, Long proved that $\sum _{k=0}^{\left(p-1\right)/2}\frac{4k+1}{{256}^k}{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^4\equiv p\left(\mathrm{mod}{p}^4\right)$ for any prime $p\ge 5$. In this paper we first reprove the above congruence by using Zeilberger's algorithm. We then give some generalizations of this supercongruence, such as $\sum _{k=0}^{\left(p-1\right)/2}\frac{(4k+1{)}^3}{{256}^k}{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^4\equiv -p\left(\mathrm{mod}{p}^4\right),$ where $p\ge 5$ is a prime of the form 3k+2. This partially confirms a recent conjecture of Guo [Some generalizations of a supercongruence of van Hamme. Integral Transforms Spec. Funct. 28 (2017), pp. 888–899].

KEYWORDS:

• Supercongruence
• binomial coefficients
• Zeilberger's algorithm

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 11A07
• 11B65
• 65B10

Acknowledgements

The author would like to thank the referee and Professor V.J.W. Guo for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author.