Finding all \(S\)-Diophantine quadruples for a fixed set of primes \(S\) (Q2232820)

Given a finite set \(S\) of primes, an \(m\)-tuple \((a_1,\ldots, a_m)\) of positive, distinct integers is called an \(S\)-Diophantine \(m\)-tuple, if for each \(1 \le i \le j \le m\), the numbers \(a_ia_j + 1\) have prime divisors coming only from the set \(S\). The author gives an algorithm to determine all \(S\)-Diophantine quadruples, supposed \(|S|=3\). It is shown that there is no \(\{2, 3, 5\}\)-Diophantine quadruple. For \(S = \{p, q,r\}\) with \(2 \le p \le q \le r \le 100\) all \(S\)-Diophantine quadruples are determined. The algorithm is implemented in Sage, the running times took a few hours on a laptop. The problem is reduced to a linear form in four logarithms of algebraic numbers. Baker-type estimates and LLL basis reduction algorithm are applied.