30 problems

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 ${\bf x}^m=\{x_1,\ldots,x_m\}$ and ${\bf y}^m=\{y_1,\ldots,y_m\}$. For an arbitrary homogeneous symmetric polynomial $S_n({\bf x}^m)$ of degree $n$, write … where…

Recurrence and coefficient conjecture. The linear recurrence corresponding to $k$ has order $\left\lfloor k/2\right\rfloor+3$, and each coefficient $c_j(q)$ is a linear polynomial…

Mod-3 power-sum conjecture. A real multiset $X$ of cardinality $n$ is uniquely determined by the $n$ values

Let $q$ be a prime power and let $n$ satisfy the normal conditions for the Restricted Problem. An element $a\in\mathbb{F}_q$ is 5-potent if it is a fifth power, and $n$-potent if i…

Let $K$ be a field of characteristic zero, let $S=K[x_1,x_2,x_3,x_4]$, and write $p_m=x_1^m+x_2^m+x_3^m+x_4^m$. Let $A=\{a_1,a_2,a_3,a_4\}$ be a set of positive integers with…

Let $K$ be a field of characteristic zero, let $p_m=x_1^m+x_2^m+x_3^m$, and let $a,b,c$ be integers satisfying $0<a<b<c$ and $\operatorname{gcd}(a,b,c)=1$. The 6abc conjecture. The…

Schäffer's conjecture. If $(k,n)\notin S$, then the equation

Kellner–Erdős–Moser conjecture. If $m\geq3$, then

Generalised Erdős–Moser conjecture. There are no integral solutions $(a,k,m)$ to

Assume $n=3$, $\gcd(a_1,a_2,a_3)=1$, and let $\mathcal{A}_p=\{a_1\bmod p,a_2\bmod p,a_3\bmod p\}$ for $p=2,3$. Let $\zeta$ be a primitive cube root of unity, and let…

Let $\tau=(\tau_1,\ldots,\tau_n)\in\mathbb{R}^n_{>0}$, and define … Let $\operatorname{Stab}(\tau)$ be the subgroup of $S_n$ consisting of coordinate permutations that fix $\tau$.…

Generic-injectivity conjecture. The recovery of a set of $n$ complex numbers from $n+1$ power sums with coprime powers is unique: for generic $z\in\mathbb{C}^n$, the fiber…

Let $S_k(n)=\sum_{j=1}^{n-1}j^k$ for positive integers $k$ and $n$. Kellner's conjecture. Let $k\ge 1$ and $n\ge 3$ be integers. Then the ratio … is an integer if and only if $n=3$…

Equality conjecture for power-sum ratios.

Conjecture on odd power sums.

Polynomiality conjecture. For the indicated integers $k$ and $j$, $S_{k,j}(n)$ is expressed as a polynomial in

Polynomial-factorization conjecture. There exist polynomials $P_k(t,x)$ in $x$ of degree $k$, with coefficients rationally depending on $t$, such that

The generalized-sums equality. For any $k\geq 0$ and $j\geq 1$, one has

Fortuny–Grau–Oller-Marcén–Rúa conjecture. All such power sums vanish unless all of the following conditions hold: $d=2$; $|R|\equiv 2\pmod 4$ and $1<k\equiv 0,\pm1\pmod 6$; and the…

Let $R$ be a finite commutative unital ring, and let $|R|$ denote its cardinality. Generalized Giuga conjecture. … if and only if $R$ is a field. This generalizes the classical Giu…

Let $d>1$ and let $R$ be a finite commutative ring. For $k\geq 1$, write $S_k(\mathbb{M}_d(R))=\sum_{A\in\mathbb{M}_d(R)}A^k$. The unique element $e\in R\setminus\\{0\\}$ such that…

Finiteness conjecture. For every $m\in\mathbb{N}$, the set of solutions to the congruence

Melham's conjecture. For any positive integers $n,m$, there is a polynomial $P_{2m-1}(x)$ of degree $2m-1$ with integer coefficients such that

Conca–Krattenthaler–Watanabe's conjecture. The polynomials $p_a(3),p_b(3),p_c(3)$ form a regular sequence in $S$. Conca et al. verified some special cases, but the conjecture remai…