35 problems

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…

Let $A_i$ and $B_j$ be Sakai's $W(E_8)$-invariant Jacobi forms, let $E_6$ be the classical modular form, and let $P_{16,5}$ be the exceptional polynomial … A Jacobi form expressed…

Let $n$ range over the positive integers, and write $\binom{2n}{n}$ for the central binomial coefficient. Pomerance's conjecture. The divisibility condition … holds on a set of pos…

Sun's conjecture. If

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

Gandhi's conjecture. For every $n\in\mathbb N$, the integer $2^n$ divides the numerator of the rational number $C_{2n}$. Carlitz proved this conjecture in 1965 using properties of…

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…

Asymptotic estimates conjecture. For all $a>0$, the estimates

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 $\mathcal{D}_{[n]}$ be the divisibility poset on $[n]=\{1,\ldots,n\}$, and let $\dim \mathcal{D}_{[n]}$ denote its order dimension. Let $\dim_2 \mathcal{D}_{[n]}$ denote the mi…

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 $\mathcal{D}_{[n]}$ be the divisibility poset on $[n]$. Write $\dim(\mathcal{D}_{[n]})$ for its order dimension, and let $\dim_2(\mathcal{D}_{[n]})$ be its 2-dimension, the sma…

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 $m,n\in\mathbb{Z}^+$, let $\pi(x)$ denote the number of primes not exceeding $x$, and let $p_r$ denote the $r$-th prime. Divisibility conjecture. For every positive integer $m$…

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…