177 problems

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 $a(m)$ 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…

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 $c(n)$ 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 $N=16n+6$ 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…

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 $p$ 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…

Fix an integer $d\geq2$. For a monic polynomial $f(x)\in\mathbb Z[x]$, consider integers $b$ such that whenever $p^2$ divides $b$, the prime $p$ does not divide the discriminant of…

Let $d\geq1$ be an integer. For every prime power $q$, let $\Omega(q)\subset\mathbb Z/q\mathbb Z$ contain at most $d$ residues, and define $\Omega(b)$ for general $b$ by the Chines…

For integers $a$ and $b$, consider positive solutions of $x^2\equiv a\pmod b$. The congruence-spacing conjecture. There exists a constant $N$ such that, for every $a$ and $b$, ther…

Integral-closure conjecture. If $k<p-1$, then $\mathbf{T}^{+}$ and $\mathbf{T}^{-}$ are integrally closed. Equivalently, every congruence between distinct eigenforms in…

Let $\Gamma$ be a noncongruence subgroup of $SL_2(\mathbb Z)$ with finite index, let $k\geq 2$, and let $S_k(\Gamma)$ have dimension $d$. Denote by $X_{\Gamma}$ the modular curve a…

Let $\mathcal{P}(n)$ be the partition function, and define … Let $\mathbb{B}_n$ be the complete Bell polynomial, and define the alternating harmonic number … For the polynomials…

Let $n>1$ and $r>0$ be integers. Define $s_r(n)$ as the least positive integer $k$ such that every sequence of $k$ integer vectors in the relevant rank-$r$ setting contains a selec…

Let $n>1$ and $r>0$ be integers. Define $w_r(n)$ as the least positive integer $k$ such that, for any $k$ integer vectors ${\bf a}_i=(a_{i1},\ldots,a_{ir})$ not congruent to…

The coefficient characterization conjecture. Under this assumption, $A\equiv0\pmod4$ and $B\equiv2\pmod4$.

Let $a,b,c$ be non-zero integers with $a,b,c$ setwise coprime, and let $d\geqslant 1$ be another integer. For a prime $q\geqslant 3$, define $N_{a,b,c,d}(L,M,N;q)$ to count triples…

For a positive integer $n$, let $\overline{p}(n)$ denote the number of overpartitions of $n$, and let $\left(\frac{n}{p}\right)$ denote the Legendre symbol for an odd prime $p$. Co…

Let $\mathcal{L},\mathcal{L}'\in\mathbb{Q}_p$ be admissible, let $k$ be the weight, and let $C_{p,k}$ be the constant appearing in the conjecture. Let…