Two-parameter families of uniquely extendable Diophantine triples

TL;DR

This paper proves that specific families of Diophantine triples can be uniquely extended to quadruples, providing new insights into the structure and extension properties of these number sets.

Contribution

It establishes the unique extension property for two-parameter families of Diophantine triples to quadruples, a novel result in the study of Diophantine equations.

Findings

Each D(u^2)-triple in the family has a unique extension to a D(u^2)-quadruple.

The proof applies to triples with parameters K and A, and u in {-2,-1,1,2}.

The result advances understanding of the extension behavior of Diophantine triples.

Abstract

Let A,K be positive integers and u=-2,-1,1 or 2. The main contribution of the paper is a proof that each of the D(u^2)-triples {K,A^2K+2uA,(A+1)^2K+2u(A+1)} has unique extension to a D(u^2)-quadruple.