A note on the number of \(S\)-Diophantine quadruples (Q2925404)

Let \(S\) be a fixed, finite set of primes. We call an \(m\)-tuple \((a_1,\ldots,a_m)\) of positive, pairwise distinct, integers \(S\)-Diophantine if for all \(1 \leq i < j \leq m\) the set of prime divisors of \(a_ia_j + 1\) is contained in \(S\).NEWLINENEWLINELet \(\Gamma\) be a multiplicative subgroup of \(Q^*\) of rank \(r\). Denote by \(A(n; r)\) an upper bound for the number of non-degenerate solutions \((x_1,\ldots,x_n)\in\Gamma^n\) to the linear \(S\)-unit equation NEWLINE\[NEWLINEa_1x_1 + \ldots + a_nx_n = 1 NEWLINE\]NEWLINE for any given \(a_1,\ldots,a_n\in Q^*\).NEWLINENEWLINEThe authors show that there exist at most NEWLINE\[NEWLINE(A(5; r) + A(2; r)^2)A(3; r)NEWLINE\]NEWLINE \(S\)-Diophantine quadruples. Using upper bounds for the numbers of solutions of \(S\)-unit equations they prove that there are at most NEWLINE\[NEWLINE\exp(27398 + 5136r)NEWLINE\]NEWLINE \(S\)-Diophantine quadruples.