9 problems

Swisher's general VanHamme-type supercongruence conjecture (F.3).

Guo–Liu–Schlosser conjecture. Under the respective hypotheses above, the corresponding congruence holds. The modulus-$p^5$ restriction of the first congruence was confirmed by Guo,…

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…

Deines et al.'s supercongruence. The congruence

Let $r\leqslant 1$ be an integer coprime with $3$. Let $p\geqslant 7$ be a prime such that $p\equiv r\pmod{3}$ and $p\geqslant 3-2r$. Write $(a)_k=a(a+1)\cdots(a+k-1)$ for the risi…

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 $p>2$ be prime. For each nonnegative integer $k$, let $C_k=\frac{1}{k+1}\binom{2k}{k}$ be the $k$-th Catalan number. Let $\Gamma_p$ denote Morita's $p$-adic Gamma function, and…

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…