154 problems

Beiter's conjecture. Sister Marion Beiter conjectured that

Let $A$ be a finite set of integers, and write its mask polynomial as $A(X)=\sum_{a\in A}X^a$. Let $S_A$ be the set of prime powers $p^\alpha$ such that $\Phi_{p^\alpha}(X)$ divide…

Corrected Beiter conjecture. Moree and Gallot proposed that

Schur's finiteness conjecture. There are only finitely many pairs $(a,p)$ with $a<0$ such that $P_{p,a}(X)$ has no linear factor and is not irreducible.

Let $p$ and $q$ be any distinct primes, and let $\Phi_n(x)$ denote the $n$th cyclotomic polynomial. Feit–Thompson conjecture. … This conjecture concerns a divisibility relation bet…

For any complex numbers $x$ and $q$, define the $q$-shifted factorial by … and, for $n\in\mathbb{Z}^{+}$, … Write $(x_1,x_2,\dots,x_m;q)_n=(x_1;q)_n\cdots(x_m;q)_n$, let…

Positive-exponent bound conjecture. The positive exponent indices satisfy

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

Let $\mathcal{P}$ be the positive integers equipped with the relation $\prec$ defined by $m\prec n$ if $\Phi_m(q)\leqslant\Phi_n(q)$ for every integer $q\geqslant 2$ and $m\ne n$.…

Let $\mathcal{P}$ be the set of positive integers, and let $\Phi_d(t)$ be the $d$th cyclotomic polynomial. Define a relation $\preceq$ on $\mathcal{P}$ by … and define $m\prec n$ w…

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 $n$ 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…

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 ……

Short-interval prime conjecture. The interval in question should contain a prime $p$. The existence of such primes would make the conditional construction of ternary cyclotomic pol…