25 problems

Let $\overline{h}_k(x)=\nu_k\,p_{k,1}(x)\,p_{k,2}(x)\cdots p_{k,\alpha}(x)=\nu_k\,\widetilde{p}_k(x)$, where each $p_{k,n}$ is monic and $\nu_k$ is rational. Write the OEIS sequenc…

The 6n divisibility conjecture. For all positive $r$, $s$, $t$, and $n$, $S_{6}(r,s,t;n)$ is divisible by both

Let $a$ and $b$ be positive integers with $a\mid b$, and let $S$ be a finite gcd-closed set of positive integers. For each $x\in S$, let $G_S(x)$ denote the set of greatest-type di…

Size-bound conjecture. For any integer $n>1$,

Let $G_{k,D}^{(r)}$ be the CM higher-order-pole forms, let $k\in\{4,6,8,10,14\}$, $1\leq r\leq k-1$, and let $D<-4$ be a discriminant with class number $1$. The complementary-pole…

Let $G_{k,D}^{(r)}$ denote the integral linear combination of $E_k/(j-j(\alpha_D))^i$ for $1\leq i\leq r$ having a pole of order $r$ at the CM point of discriminant $D$. Let…

Let $n,m$ be positive integers and let $e\in\mathbb{Z}^n$ and $f\in\mathbb{Z}^m$ be arbitrary. Write $Z_{\mathrm{quotient}}^c(q,t)$ for the quotient generating series, and interpre…

Let $G$ be a finite group, let $G'$ denote its commutator subgroup, and let $h(G)$ be the Harada number … where $K_1,\ldots,K_s$ are the conjugacy classes of $G$ and…

Let $\mathfrak{g}$ be a Lie algebra with Coxeter number $h$, let $\mathfrak{u}_q(\mathfrak{g})$ be the corresponding small quantum group, and let $L$ be an irreducible non-trivial…

For $0<\alpha<1$, let $\mathcal{D}_{(\alpha n,n]}$ be the divisibility poset on the integers in $(\alpha n,n]$. For an integer $t\geq2$, let $\dim_t(P)$ denote the $t$-dimension of…

Let $a$ be an even integer and $b$ an odd integer, and let $s_{\sqrt{-1}}(a;b)$ and $Q(a;b)$ denote the elliptic Dedekind sum and the associated quantity used in the paper. For an…

Let $L=(1,-1)$, let $K$ be the parameter appearing in the definition of the generalized sequence $b_T$, and let $h\in\mathbb{N}$. Infinitely-many-instances conjecture for $b_T$. If…

Let $L=(1,-1)$ and $R=(1)$, and let $K$ be the parameter appearing in the definition of the generalized sequence $b_T$. Write $\operatorname{\underline{\deg}} K$ for its indicated d…

Let $b_{2}(n)$ denote the number of binary partitions of $n$. Alkauskas's conjecture. For every $h\in\mathbb{N}$ with $8\nmid h$, there exist infinitely many $n$ such that $h$ divi…

Let $p$ be prime, let $e<f$ be positive integers, and set $m=p^e$, $n=p^f$, and $k=n/m=p^{f-e}$. Let $M^{se}$ and $M^{es}$ denote the change-of-basis matrices between the Schur and…

Let $p$ be prime, let $e<f$ be positive integers, and set $m=p^e$, $n=p^f$ as above. With $n=km$, consider the symmetric-function expression … where the sum runs over partitions fi…

Wu's divisibility-by-3 conjecture. If $n$ is not a multiple of $11$ with an even number of digits and $a(n)=0$, then $n$ is a multiple of $3$.

Let $a,b$ and $n$ be positive integers with $a>b$. For integers $x$ and $y$, write $x\mid y$ when $x$ divides $y$. Amdeberhan–Moll's divisibility conjecture. … The conjecture conce…

Let $a,b$ and $n$ be positive integers with $a>b$. Amdeberhan's divisibility conjecture. … This is a generalization of a theorem in the paper, found by T. Amdeberhan in personal co…

Divisibility by six conjecture. For every $n,N\geq 3$, $a_{n,N}$ is divisible by $6$.

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper. For positive integers $r,s,t$ satisfying $r+s+t\equiv 1\pmod 2$, triple-product divisibility conjecture. The…

Let $B_{m,k}$ and $B_{n,k}$ denote the Catalan-triangle quantities used in the paper. Let $m,n,s,t\geqslant 1$ and $r\geqslant 0$, with $r+s+t\equiv 1\pmod 2$. Mixed-product divisi…

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper, and let $n,r\geqslant 1$. Odd-power Catalan-triangle congruence conjecture. … if and only if $n=2^a-2^b$ for…

Let $n,r\geqslant 1$. Define $\alpha(N)$ to be the number of $1$'s in the binary expansion of $N$. Binary-weight divisibility conjecture. The sum … is divisible by … The conjecture…

Almost-divisibility conjecture. For every $\epsilon>0$, one can take $L=X^\epsilon$ provided $X$ is sufficiently large in terms of $\epsilon$.