The extensibility of the Diophantine triple $\{2, b, c\}$

TL;DR

This paper investigates the conditions under which Diophantine triples of the form {2, b, c} can be extended to quadruples, proving non-extendability for certain families and characterizing regular quadruples.

Contribution

It demonstrates that certain Diophantine triples cannot be extended to irregular quadruples and characterizes when quadruples are necessarily regular based on properties of b.

Findings

Certain families of c prevent extension to irregular quadruples

All quadruples with b/2-1 prime are regular

Extension to irregular quadruples is generally impossible for these sets

Abstract

The aim of this paper is to consider the extensibility of the Diophantine triple $\{2,b,c\}$, where $2<b<c$, and to prove that such a set cannot be extended to an irregular Diophantine quadruple. We succeed in that for some families of $c$'s (depending on $b$). As corollary, for example, we prove that for $b/2-1$ prime, all Diophantine quadruples $\{2,b,c,d\}$ with $2<b<c<d$ are regular.