5 problems

Let $r\leq 1$ be an odd integer coprime with $5$, and let $p$ be an odd prime such that $p\equiv r\pmod{5}$ and $p\geq (5-3r)/2$. Then the proposed supercongruence. … This is prese…

Let $p$ be an odd prime. For parameters $a_1,,a_6$ and $b_1,,b_5$, write the truncated hypergeometric series as … where $(a)_0=1$ and $(a)_k=a(a+1)\cdots(a+k-1)$ for $k\geq 1$. Let…

Let $p\geq 5$ be a prime satisfying $p\equiv 3\pmod{4}$, and let $\Gamma_p$ denote the $p$-adic Gamma function. A modulo $p^5$ extension of van Hamme's (A.2) supercongruence. … Thi…

Let $p\ge 5$ be a prime. For a $p$-adic integer $a$, let $\langle a\rangle_p$ denote its least nonnegative residue modulo $p$, and let $\Gamma_p$ denote the $p$-adic Gamma function…

Let $p\ge 5$ be a prime. For a rational number $x$ and a positive integer $n$, write … for the truncated hypergeometric series, where $(a)_k$ is the rising factorial, and let…