On a Supercongruence Conjecture of Z.-W. Sun

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) $\sum\limits_{k = 0}^{\left\lfloor {{5 \over 6}{p^a}} \right\rfloor } {{{\left( {\matrix{{2k} \cr k \cr } } \right)} \over {{{16}^k}}} \equiv \left( {{3 \over {{p^a}}}} \right)\,\,\left( {\bmod \,{p^2}} \right)} $ is obtained, where (÷) is the Jacobi symbol.

Keywords

Supercongruences / Binomial coefficients / Fermat quotient / Jacobi symbol

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