81 problems

Let $n$ be a positive odd integer with $n\equiv 3\pmod{4}$, let $q$ be an indeterminate, and let $(a;q)_k$ denote the $q$-shifted factorial. Write…

Let $p\ge7$ be a prime with $p\equiv3\pmod4$, and write $n=r(p-1)$ with $r\ge1$. Odd-multiple valuation conjecture. … Equivalently, among multiples of $p-1$, … for $p\equiv3\pmod4$…

Let $p$ be an odd prime and define … with $v_p(x+yi):=\min\{v_p(x),v_p(y)\}$. The mod-4 valuation conjecture. For $1\le n\le p-2$, … and for $4\le n\le p-2$, … These formulas are s…

Let $C/\mathbb{Q}$ be an elliptic curve with $j(C)\notin\{0,1728\}$. A prime $p$ is supersingular for $C$ when $a_p(C)=0$ (for $p\geq5$), where $a_p(C)=p+1-|C(\mathbb{F}_p)|$. Let…

Let $n\equiv 1\pmod{4}$ be a positive integer. Define the $q$-integer $[m]=(1-q^m)/(1-q)$, the $q$-shifted factorial $(x;q)_k=(1-x)(1-xq)\cdots(1-xq^{k-1})$, and let $\Phi_n(q)$ de…

Product supercongruence conjecture. For primes $p\geq 5$,

Let $p\ge 5$ be a prime, and let $\\{u_n\\}_{n\ge 0}$ be one of the sequences $\mathbf{B}$ and $\mathbf{F}$, defined by … or … Here $\left(\frac{p}{3}\right)$ is the Legendre symbo…

For every prime $p\ne3$, consider the truncated hypergeometric sum … Let $a_p$ be the coefficient of $q^p$ in the weight $4$ cusp form of level $54$ beginning…

For the Hulek–Verrill family $1-tg(\boldsymbol{x})=0$, define … and … Let $a_p$ be the coefficient of $q^p$ in the weight $4$ cusp form of level $14$ beginning $q-2q^2+8q^3+\cdots$…

Two-term supercongruence conjecture. For all primes $p\geq 5$ and all integers $m,r\geq 1$,

Let $m$ and $n$ be positive integers with $n$ odd, and let $(a;q)_k$ denote the q-shifted factorial. The cubic q-supercongruence conjecture. … The source records this as another si…

Let $m$ and $n$ be positive integers with $n$ odd, and let $(a;q)_k$ denote the q-shifted factorial. The cubic q-supercongruence conjecture. … The $m=1$ case was proved by the auth…

Let $m$ and $n$ be positive integers with $n\equiv1\;\allowbreak(\operatorname{mod}4)$, and let $(a;q)_k$ denote the q-shifted factorial. The fifth-order q-supercongruence conjectu…

Let $m$ and $n$ be positive integers with $n\equiv1\;\allowbreak(\operatorname{mod}4)$ and $n>1$, and let $(a;q)_k$ denote the $q$-shifted factorial. The q-analogue conjecture. … T…

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 $(r_1,r_2,q)$ range over … Set $\mathbf{\alpha}_{(r_1,r_2)}=\{r_1,1-r_1,r_2,1-r_2\}$ and $\mathbf{\beta}_q=\{1,1,q,2-q\}$. Let $\mathrm{HD}_{(r_1,r_2,q)}$ denote the hypergeome…

Let $(x)_k$ denote the rising factorial, with $(x)_0=1$. Let $d\geqslant 2$ be an even integer, and let $p\equiv d+1\pmod{2d}$ be a prime. The hypergeometric supercongruence. … Thi…

Let $p>3$ be a prime with $p\equiv3\pmod4$, and let $r$ be a positive integer. The supercongruence. … This is described as a generalization of an earlier case distinction; the stat…

The supercongruence.

Triple-sum Ramanujan-type congruence. One has

Let $F_N(t)=\sum_{n=0}^{N-1}f_nt^n$ be the truncation of $F(t)$ at $t^N$, and let $t\mapsto t^\sigma$ be the excellent Frobenius lift. The congruence … holds for every $m,s\geq 1$.…

Let $p\equiv 5\pmod{8}$ be a prime, and let $(a)_k=a(a+1)\cdots(a+k-1)$ denote the rising factorial. The p-adic supercongruence conjecture. … This is presented as a stronger versio…

Bernoulli congruences for $R_8^{(m)}(p)$, $R_9^{(m)}(p)$, and $R_{10}^{(m)}(p)$.

Let $\boldsymbol{k}$ be an index, let $\operatorname{wt}(\boldsymbol{k})$ denote its weight, and let $\zeta_{\widehat{\mathcal{A}}}(\boldsymbol{k})$ and…

Let $k$ be a positive integer and let $p_1$ be a prime greater than $k+1$. For each index $\boldsymbol{k}$, let $H_{p-1}(\boldsymbol{k})$ be the multiple harmonic sum, and let…