16 problems

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

Uniqueness conjecture for extensions. Then

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 $\mathcal{R}$ 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…

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

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

Classification conjecture. One has

Let $p<q$ be primes. A pair $(p,q)$ is called a Wieferich pair when it satisfies the paper's Wieferich condition. Wieferich-free two-prime $S$-Diophantine quadruple conjecture. If…

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 $p<q$ be primes. A pair $(p,q)$ is called a Wieferich pair when it satisfies the paper's Wieferich condition. Wieferich-free two-prime $S$-Diophantine quadruple conjecture. If…

Let $p<q$ be primes and let $S=\{p,q\}$. An $S$-Diophantine quadruple is a quadruple of distinct positive integers such that every pairwise product plus one has all prime divisors…

Dujella's conjecture. For every $n\in S$, there does not exist a $D(n)$-quadruple of natural numbers.

Unique extension conjecture. Then $d=d_{+}$.

The regularity conjecture for Diophantine quadruples. If $d>\max\{a,b,c\}$, then $d=d_+$. All known Diophantine quadruples were regular when this conjecture was formulated, and the…

Stronger Diophantine quadruple conjecture. If $\{a,b,c,d\}$ is a Diophantine quadruple and $d>\max\{a,b,c\}$, then $d=d_+$.

Szalay–Ziegler conjecture. No $\{p,q\}$-Diophantine quadruple exists.