48 problems

Erdős–Moser conjecture. The equation

Schäffer's conjecture. If $(k,n)\notin E$, then the only nontrivial integer solution of

For each $k\in\mathbb{N}$, let $g(k)$ be the smallest number such that every positive integer is a sum of $g(k)$ non-negative $k$-th powers. Waring's conjectural formula. It is con…

Let $r,s,k$ be positive integers. Suppose that … where $a_1,\ldots,a_r,b_1,\ldots,b_s$ are positive integers satisfying $a_i\neq b_j$ for every $1\leq i\leq r$ and $1\leq j\leq s$.…

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 $n=3$…

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…

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

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

Let $m$ be odd and let … Define … and let $S_m^{(r)}(n)$ denote the higher-order power sums used in the source. Svinin's multiple-sum conjecture. The sum…

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 $F_n$ denote the $n$th Fibonacci number, defined by … where $\alpha$ and $\beta$ are the roots of $t^2-t-1$. For positive integers $n,k,p,q$, consider … The solutions…

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…

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 $S(n,k)=\sum_{i=1}^{n}i^k$ denote the sum of the $k$-th powers of the first $n$ positive integers. Let $k\geq 1$ and $m\geq 3$ be positive integers. Erdős–Moser equation conjec…

Set $S=\mathbb{C}[x_1,x_2,x_3,x_4]$, and for every positive integer $m$ let $p_m=x_1^m+x_2^m+x_3^m+x_4^m$. The $(p_2,p_b)$ normality conjecture. The quotient ring $S/(p_2,p_b)$ is…

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…

Conjecture 3.1. For all positive integers $x$ and $n$,

Let $R(k,m)$ be defined by … and let $S_{2k}(n)$ denote the sum of the $2k$th powers of the relevant integers, as in the preceding formulas. The entries in the $k$th row of the tri…

Consider the homogeneous system (HS) associated with the three-variable power-sum system, with $n=3$ and $\gcd(a_1,a_2,a_3)=1$, and let $i_{\mathcal{A}}$ be the nilpotency index ap…

Let $n=3$, $\gcd(a_1,a_2,a_3)=1$, and set $\mathcal{A}_p=\{a_1\bmod p,a_2\bmod p,a_3\bmod p\}$ for $p=2,3$. Let $i_{\mathcal{A}}$ be the index of nilpotency of the zero-divisor…

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…