Diophantine $$\\mathcal{S} $$S-quadruples with two primes which are twin

Abstract

We show that there are only finitely many pairs of twin primes $$(p, p+2) $$(p,p+2) such that there exists an$$\\mathcal{S} $$S-Diophantine quadruple in the sense of Szalay and Ziegler for the set$$\\mathcal{S} $$S of integers composed only of primes p and p + 2.