On the Diophantine pair \(\{a,3a\}\) (Q2039520)

A set of positive integers is called Diophatine if the product of any two distinct elements is one less than a perfect square. It is known that if the set \(\{a,b,c,d\}\) is Diophantine and \(a<b\le 8a\), \(b<c<d\), \(d>4c(ab+1)\), then \(c\) must be one of four values given explicitly in terms of \(a\) and \(b\). By using standard tools in this kind of problems, the authors prove that if \(b=3a\) or \(b=8a\), then none of the admissible values for \(c\) can be attained. Therefore, any Diophantine quadruple that contains either \(\{a,3a\}\) or \(\{a,8a\}\) must be regular.