Diophantine quintuples containing triples of the first kind

Abstract

We consider Diophantine quintuples $\{a,b,c,d,e\}$, sets of distinct positive integers the product of any two elements of which is one less than a perfect square. Triples of the first kind are the subsets $\{a,b,d\}$ with $d>b^5$. We show that there are no Diophantine quintuples containing triples of the first kind.

10 pages