On the existence of $S$-Diophantine quadruples

Abstract

Let $S$ be a set of primes. We call an $m$-tuple $(a_1,\dots ,a_m)$ of distinct, positive integers $S$-Diophantine, if for all $i\ne j$ the integers $s_{i,j}:=a_i{a}_j+1$ have only prime divisors coming from the set $S$, i.e. if all $s_{i,j}$ are $S$-units. In this paper, we show that no $S$-Diophantine quadruple (i.e.~$m=4$) exists if $S=\{3,q\}$. Furthermore we show that for all pairs of primes $(p,q)$ with $p<q$ and $p\equiv 3\mathrm{mod}4$ no $\{p,q\}$-Diophantine quadruples exist, provided that $(p,q)$ is not a Wieferich prime pair.

References in corpus (1)

• A note on the paper by Bugeaud and Laurent "Minoration effective de la distance p-adique entre puissances de nombres algébriques"