102 problems

Coprime dice relabeling conjecture. There are no ways to relabel a die of size $r$ and a die of size $s$ without changing the frequencies of their sums.

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…

For odd primes $p<q<r$, let $A(n)$ denote the height of the cyclotomic polynomial $4\Phi_n(X)$. Beiter's conjecture. … for all odd primes $p<q<r$. The conjecture was disproved by L…

Corrected Beiter conjecture. Moree and Gallot proposed that

Ciolan–García-Sánchez–Moree conjecture. A numerical semigroup $S$ is cyclotomic if and only if it is complete intersection.

Let $n$ be a positive odd integer, let $(a;q)_k$ denote the $q$-shifted factorial, and let $\Phi_n(q)$ be the $n$th cyclotomic polynomial. Guo's conjecture. Modulo $\Phi_n(q)^2$, t…

For a natural number $h$, let $n_h$ be the smallest ternary integer, if it exists, such that $A(n_h)=h$. Polynomial-growth conjecture for the least ternary height representative. T…

Let $h_{105}(x)$ be the coordinator polynomial of the lattice $\mathbb Z[\zeta_{105}]$ with respect to the set of all $105$th roots of unity. A polynomial is palindromic if its coe…

Let $\zeta_{15}=e^{2\pi i/15}$, and let $h_{15}(x)$ be the coordinator polynomial of the lattice $\mathbb Z[\zeta_{15}]$ with respect to the set of all $15$th roots of unity. Parke…

Let $p$ be an odd prime, let $\zeta_{2p}=e^{2\pi i/(2p)}$, and let $h_{2p}(x)$ be the coordinator polynomial of $\mathbb Z[\zeta_{2p}]$ with respect to the set of all $2p$th roots…

Let $m$ be a positive integer. Write $\sqrt m$ for the squarefree part of $m$, let $\zeta_m=e^{2\pi i/m}$, and let $h_m(x)$ be the coordinator polynomial of the lattice…

Let $a_n(k)$ denote the coefficient of $x^k$ in the cyclotomic polynomial of order $n$, let $\delta(a_n(k)=v)$ denote the density of integers $n$ for which this coefficient equals…

Let $a_n(k)$ denote the coefficient of $x^k$ in the cyclotomic polynomial of order $n$, and let $M_n(a_n(k))$ denote the limiting mean value of $a_n(k)$ as $n$ varies. For $k\ge1$,…

Let $\mathcal{P}_3$ be the prime set used in the paper, let $\Phi_{4p}(2)$ be the cyclotomic value, and let $\omega(N)$ denote the number of distinct prime divisors of $N$. Divisor…

Let $\mathcal{P}_3$ and $\mathcal{S}_3^{(\leq 3)}$ be as used in the paper, and define … … Here $\Phi_{4p}(2)$ is the value of the cyclotomic polynomial $\Phi_{4p}$ at $2$. Divisor…

Let $\mathcal{P}_3$ be the set of odd primes used in the paper whose shifted-prime closure has the stated 3-Higgs property, and let $\mathcal{S}_3^{(\leq 3)}$ be the associated bou…

Let $\mathcal{S}_3^{(\leq 3)}$ be the bounded-exponent semigroup used in the paper, and let $\operatorname{ord}_r(2)$ denote the multiplicative order of $2$ modulo a prime $r$. Hyb…

Gatzweiler–Krattenthaler conjecture. If this fraction is a polynomial in $q$, then it has non-negative coefficients. This is presented as a strengthening of Stanton's conjecture, r…

Let $\mathbf{a}=(a_1,\ldots,a_n)$ be a sequence of non-negative integers, and let $m$ be a non-negative integer. Consider … A sequence $\mathbf{a}$ is symmetric when its entries ar…

For an odd prime $p$, define … where $q$ and $r$ range over primes. Bachman–Bao–Wu's conjecture. For every odd prime $p$ and every integer $h$ with $1\le h\le M(p)$, the equation ……

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…

Gimbert's cyclotomic irreducibility conjecture. If $k$ is even, then $F_{i,k}(x)$ is reducible in $\mathbb Q[x]$ if and only if $i\mid(k+2)$, and in that case it has exactly two fa…

Let $n$ and $s$ be the parameters occurring in Lemma 3, and let $\Phi_{n^s}(q)$ be the corresponding cyclotomic polynomial. The strengthening conjecture. The q-congruence in Lemma…

Height-realisation conjecture. For every odd prime $p$ and every integer $h$ with $1\le h\le M(p)$, there exist primes $q$ and $r$ such that

Let $n$ be a positive squarefree integer, let $d>1$ be an integer not dividing $n$, set $d_1=\gcd(d,n)$, and let … Let $F_n$ be the Fekete polynomial and let…