13 problems

Let $P_8(x)=3x^2-2x$ for an integer $x$, and for positive integers $a\leq b\leq c\leq d$ define … A sum is universal over $\mathbb Z$ if the equation … has an integer solution for…

For any $a_1,b_1,c_1\in\mathbb{Z}^+$ and $a_2,b_2,c_2\in\mathbb{Z}$ with $a_1-a_2$, $b_1-b_2$, and $c_1-c_2$ all even, consider the tuple $(a_1,a_2,b_1,b_2,c_1,c_2)$ associated wit…

Kane's conjecture. An arbitrary sum $\Delta_{\alpha_1,\dots,\alpha_k}$ represents all positive odd integers if and only if it represents each of

A ternary quadratic form is coprime-universal with respect to $p$ if it represents every positive integer coprime to $p$. Coprime-universality conjecture. The ternary forms listed…

Let $f(x,y,z,t)=x^2+2y^2+7z^2+13t^2$ be the indicated diagonal quaternary quadratic form. Halmos's conjecture. The form $f$ represents every nonnegative integer except $5$. Halmos'…

Sun's conjecture. For every $n\in\mathbb{N}$, the equation

Let $K$ be the totally real number field under consideration, let $\mathcal{O}_K^+$ denote its totally positive elements, and let $Q$ be the quadratic form discussed in the precedi…

Universality conjecture. The form is universal over $K$.

A quadratic form is called universal if it represents every positive integer. 290-conjecture. Every quadratic form, whether classical or not, is universal if it represents all posi…

Sun's conjecture. The sum $p_3+4p_4+p_5$ is universal over $\mathbb N$.

Let $a\le b\le c$ be positive integers, and for an integer $k>2$ define the generalized $k$-gonal number by … For $k=5$ or $k\ge 7$, set … A ternary sum is universal over…

For $m\geq 3$, let … be the generalized $m$-gonal number. Sun's conjecture. Every $n\in\mathbb N$ can be written in each of the forms … and … with $x,y,z\in\mathbb N$. The paper st…

Kane–Sun conjecture.