On the existence of S-Diophantine quadruples

Sa？etak Let \(S\) be a set of primes. We call an \(m\)-tuple \((a_1,\ldots,a_m)\) of distinct, positive integers \(S\)-Diophantine, if for all \(i\neq j\) the integers \(s_{i,j}:=a_ia_j+1\) 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 \(