17 problems

Regularity conjecture. Any $D(4)$-quadruple is regular.

Let $n$ be a nonnegative integer, and let $k$ be an integer. Consider the Diophantine equation … A positive integer solution means $x,y>0$, and a positive odd integer solution mean…

Let $\mathcal{A}$ be the set of positive integers $N$ for which some $d\in\mathbb{N}$ makes $N,N+d,N+2d$ a three-term arithmetic progression of consecutive powerful numbers. Let…

Let $n\neq0$ be an integer. A set of distinct positive integers is a $D(n)$-tuple if the product of any two distinct elements, increased by $n$, is a perfect square. In particular,…

Let $p\in\mathbb N$ be prime, let $K=\mathbb Q[\sqrt{p}]$, and let $R_p$ be the order of index $p$ in $K$. An order is locally associated when it has the local association property…

Let $P_3$ be the path on three vertices. For a graph $H$, a $(1,1)$-antimagic labeling is an antimagic labeling whose vertex sums are exactly $1,2,\ldots,|V(H)|$. Pell-equation exi…

Unrestricted-power four-squares conjecture. There are at most four distinct integer squares among the $y_k'$. If $|N_{\alpha}|$ is a prime power or a perfect square, then there are…

Let $a$, $b$ and $d$ be positive integers such that $d$ is not a square, let $\alpha=a+b^{2}\sqrt{d}$ have norm … and let $\varepsilon=(t+u\sqrt{d})/2$ be a unit in…

Yuan's conjecture. If this system has at least two solutions in positive integers, then its coefficients are given by

Let $d\in\mathbb{N}_{\geq 2}$ be squarefree, let $K=\mathbb{Q}(\sqrt{d})$, let $\mathcal{O}_K$ be the ring of algebraic integers of $K$, and let $\varepsilon>1$ be the fundamental…

Let $d\in\mathbb{N}_{\geq 2}$ be squarefree, let $K=\mathbb{Q}(\sqrt{d})$, and let $\mathcal{O}_K=\mathbb{Z}[\omega]$ be the ring of integers of $K$, where … Let…

Goldfeld–Hinkle's pole-free continuation conjecture. The function $L_d(s)$ has meromorphic continuation to $\textup{Re}(s)>\frac12$ with at most a simple pole at $s=\frac23$. In th…

Common-largest-elements conjecture. Then $\{a_1,a_2,b,c\}$ is a $D(4)$-quadruple.

Mordell's conjecture. The prime $p$ does not divide $y$.

Classification conjecture. One has

Fouvry's conjectural estimate. There exists an absolute constant $\vartheta\in[\frac12,1)$ such that, uniformly under these conditions,

Pell-equation lower-bound conjecture. One should have