Diophantine quadruples containing some triples and the number of Diophantine quintuples

Abstract

A set {a1,…,am} of m distinct positive integers is called a Diophantine m‐tuple if aiaj+1 is a perfect square for all i,j with 1⩽i<j⩽m. It is conjectured that all the Diophantine quadruples are regular, that is, if {a,b,c,d} with a<b<c<d is a Diophantine quadruple, then d = d+, where d+ = a+b+c+2abc+2rst and r = ab+1, s = ac+1, t = bc+1. In this note, we briefly explain how it is shown that Diophantine quadruples containing some triples are regular. We also mention a recent result on the upper bound for the number of Diophantine quintuples.