47 problems

A Diophantine quintuple in integers is a set of five distinct positive integers such that the product of any two distinct elements, increased by $1$, is a perfect square. The nonex…

Uniformity conjecture prediction. The answer to this question should be $\ell=2$. This prediction is conditional on the uniformity conjecture and is obtained by considering the dis…

A Diophantine $D(-1)$-quadruple is a set of four distinct positive integers such that the product of any two distinct elements minus $1$ is a perfect square. Dujella's conjecture.…

Let $n\ne 1$, and let a Diophantine triple with property $D(n)$ be a set of three nonzero elements $\{a,b,c\}$ such that $ab+n$, $ac+n$, and $bc+n$ are squares in the relevant comm…

Let $a,b,c$ be positive integers. Győry–Sárközy–Stewart's conjecture. The largest prime factor of … tends to infinity as $\max\{a,b,c\}\to\infty$. This conjecture concerns the prim…

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…

Let $\mathcal{R}$ be a commutative ring with unity and let $n\in\mathcal{R}\setminus\{0\}$. A Diophantine quadruple with property $D(n)$ in $\mathcal{R}$ is a set of four nonzero e…

Szalay–Ziegler conjecture. If $|S|=2$, then no $S$-Diophantine quadruple exists.

Let $q$ be a prime. The $\{5,q\}$-Diophantine quadruple conjecture. If $q\not\equiv1\pmod4$, then no $\{5,q\}$-Diophantine quadruple exists. The authors say that this should be pro…

Let a $D(1)$-quintuple be a set of five positive integers $\{a_1,a_2,a_3,a_4,a_5\}$ such that $a_i a_j+1$ is a perfect square for every $1\leq i<j\leq 5$. No-$D(1)$-quintuple conje…

For a nonzero integer $n$, let $M_n$ be the supremum of the cardinalities of sets of positive integers having property $D(n)$, where property $D(n)$ means that $a_i a_j+n$ is a per…

Let $n$ be a nonzero integer. A set of positive integers $\{a_1,\ldots,a_m\}$ has property $D(n)$ if $a_i a_j+n$ is a perfect square for every $1\leq i<j\leq m$; such a set is call…

Let a Diophantine $m$-tuple be a set $\{x_1,x_2,\ldots,x_m\}\subseteq\mathbb{Z}^+$ of $m$ distinct positive integers such that $x_i x_j+1$ is a perfect square for every…

Let $\mathbb{Z}[\omega]$ be the ring considered in the paper, and call a set of four distinct elements a $D(-1)$-quadruple if the product of any two distinct elements, decreased by…

Uniqueness conjecture for extensions. Then

Folklore conjecture. There cannot be five elements in a Diophantine tuple; equivalently, $m<5$.

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,…

A Diophantine $m$-tuple is a set of $m$ positive integers such that the product of any two distinct elements is one less than a square. The nonexistence conjecture for Diophantine…

The non-existence conjecture for Diophantine $5$-tuples. There is no Diophantine $5$-tuple.

Smallest-element uniqueness conjecture. Then $\{a_2,b,c,d\}$ is not a $D(4)$-quadruple for any integer $a_2$ with $a_1\neq a_2<b$.

Uniqueness conjecture. Any $D(4)$-triple $\{a,b,c\}$ has a unique extension to a Diophantine quadruple $\{a,b,c,d\}$ by an element $d>\max\{a,b,c\}$.

Let $\mathcal{R}$ be a commutative ring with unity, and let $n\in\mathcal{R}$. A Diophantine quadruple with property $D(n)$ is a set…

Let a Diophantine quadruple be a set of four distinct positive integers such that the product of any two distinct elements, increased by $1$, is a perfect square. Suppose that … is…

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.