177 problems

Let $t_n$ and $t_n'$ be the quantities defined earlier in the paper, and let $\left(\frac{\cdot}{p}\right)$ denote the Legendre symbol. The tangent permanent conjecture. (i) For ev…

Supercongruence conjecture modulo $p^3$. Under these conditions,

Let $c(n)$ denote the coefficient sequence under consideration, and let $n\in\mathbb{N}_0$. The congruence conjecture for $c(32n+23)$. … This conjecture is proposed as an open ques…

Let $\overline{p}_{9}(n)$ denote the $9$-color overpartition function, and let $n\ge0$ and $\alpha\ge1$. Arithmetic congruence conjecture. For all $n\ge0$ and $\alpha\ge1$, … These…

Let $\overline{p}_{6}(n)$ and $\overline{p}_{8}(n)$ denote the $2$-color overpartition functions with indicated color parameters, and let $n\ge0$. Higher-modulus congruence conject…

Let $c(n)$ denote the coefficient sequence under consideration, and let $k,n\in\mathbb{N}_0$. The infinite-family congruence conjecture for $c(n)$. … … … The authors propose these…

Let $N=16n+6$ for some nonnegative integer $n$. Define $\a0\sigma_0(m)$ to be the number of positive divisors of $m$. The $N=16n+6$ congruence conjecture. … The authors report comp…

Let $S_m(n)=\sum_{j=1}^{n-1}j^m$, and let $k$ be a positive integer. Even-modulus power-sum conjecture. For every $k\ge 1$ and every even positive integer $n$, the congruence … hol…

Let $\overline{p}_{\ell}(n)$ denote the $2$-color overpartition function, where $\ell$ is a positive integer, and let $n\ge0$. The mod 8 congruence conjecture. For all…

Let $G$ be a finite simple group of Lie type, let $p$ be a prime different from the defining characteristic of $G$, and let $m$ be a positive integer. For elements $g,g'\in G$, wri…

Congruences for Domb numbers and Lucas sequences. For every odd prime $p$: (i) if one of the Legendre symbols $(\frac{p}{3})$ and $(\frac{p}{13})$ is $1$, then

Let $a(m)$ be defined by the paper's generating function, and suppose that $m$ is a positive integer such that … Write the prime factorization of $4m+3$ as … where each…

Keith's conjecture. For every integer $n\geq 0$,

Let $C_{n,k}:=\frac{n-2k}{n}{n\choose k}={n-1\choose k}-{n-1\choose k-1}$, with the Catalan triangle relation $B_{n,k}=C_{2n,n-k}$. Let $n$ and $a$ be positive integers with $n>a$,…

For $n\in\mathbb{N}$, let … A prime $p$ satisfying $w_p\equiv 1\pmod{p^3}$ is considered in the context of Wolstenholme's theorem. Jones' conjecture. … This asserts that the known…

Integrality conjecture. Then

Let $m$ and $n$ be in the critical range for the algebraic Hecke character $\psi_m$, and define … Suppose a prime $\ell>m$ divides the denominator of $Q(m,n)$. Harder's conjecture.…

Let $\overline{OPT}_k(n)$ denote the number of overpartition $k$-tuples of $n$ with odd parts. Das–Saikia–Sarma's second conjecture. For all $i\geq 1$, all $n\geq 0$, and integers…

Let $\overline{p}_{q}(n)$ denote the number of overpartition $q$-tuples of $n$, where $q$ is prime. Saikia's conjecture. For all $n\geq 0$ and primes $q$, … These congruences exten…

Wolstenholme's higher-power congruences. For every such $k$ and $p$,

Let $S_k$ and $S_k^{(2)}$ be the sequences defined by the corresponding central-binomial sums in the source. Sun's integrality conjectures. For every positive integer $n$, one has…

Let $p$ be a prime with $p\geq 5$. Define the $q$-analogue of the trinomial coefficient by … where $[n]_q=(1-q^n)/(1-q)$ and $\lfloor x\rfloor$ denotes the integral part of the rea…

Let $\theta$ satisfy $1/2<\theta\leq 1$. Let $q$ be a positive integer and let $c$ be an integer with $(c,q)=1$. Coprime-factor congruence conjecture in short intervals. There is a…

Let $\epsilon>0$ be any small positive real number, let $q$ be a positive integer, and let $c$ be an integer with $(c,q)=1$. Short-factor congruence conjecture. The congruence … ha…

Let $L(k\Lambda_0)$ denote the level-$k$ vacuum module, let $\mathcal{F}_{L(k\Lambda_0)}(q)$ be its normalized generating function, and let $p=2k+3$ be a prime with $p\geq 5$. Cong…