S-Diophantine quadruples with $\mathbf{S}=\{2,\mathbf{q}\}$

Abstract

Let $S$ denote a set of primes and let $a_1,\dots ,a_m$ be positive distinct integers. We call the $m$-tuple $(a_1,\dots ,a_m)$ an $S$-Diophantine tuple if $a_i{a}_j+1=s_{i,j}$ are $S$-integers for all $i\ne j$. In this paper, we show that no $S$-Diophantine quadruple (i.e~$m=4$) exists if $S=\{2,q\}$ with $q\equiv 3(\mathrm{mod}4)$ or $q<10^9$. For two arbitrary primes $p,q<10^5$ we gain the same result.