33 problems

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

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

Let a $D(-1)$-quadruple be a set of four non-zero integers such that, for every two distinct elements $a$ and $b$, the number $ab-1$ is a perfect square. Nonexistence conjecture fo…

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

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

Cohn's even-exponent conjecture. If $n$ is even, then every solution satisfies either $(n,a,b)=(4,13,239)$ or $n=2$. The source presents this as a conjecture for the even-exponent…

Let $d$ range over discriminants satisfying the Pellian condition, namely those for which every odd prime divisor of $d$ is congruent to $1$ modulo $4$. Consider the discriminants…

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 $d\ge 1$ be squarefree, and consider the negative Pell equation … The source distinguishes rational solutions, for which the Hasse principle gives a characterization, from inte…

Let $d$ range over positive integers, and let $\varepsilon_d$ denote the fundamental solution or fundamental unit associated with the Pell equation of discriminant parameter $d$. H…

Let $D$ be a positive nonsquare integer. Consider the positive integer solutions of the Brahmagupta–Pell equation … Let $(u_1,v_1)$ be the least positive integer solution, and 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…

Goldfeld–Hinkle's pole conjecture. The latter case does not occur: $L_d(s)$ has no other poles with $\textup{Re}(s)>\frac12$ besides the possible simple pole at $s=\frac23$.

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

The irregular D(4)-quadruple conjecture. An irregular $D(4)$-quadruple does not exist.

Stevenhagen's conjecture. Based on his heuristic model,

Let $\mathcal{D}$ be the set of squarefree integers $d>0$ such that every prime divisor $p$ of $d$ satisfies $p\equiv 1,2\pmod 4$, and let $\mathcal{D}^-$ be the set of squarefree…

Dujella's conjecture. The equation has at most one positive solution satisfying

Let $\mathbb{B}_n$ be the $n$-th layer of the $\mathbb{Z}_2$-extension of $\mathbb{Q}$, and let $u_n\in\mathbb{B}_n$ be the explicit unit obtained from the paper's generalized Pell…

Classification conjecture. One has

Nonexistence conjecture. If