# Extendibility of Some P_{k} Sets

## TL;DR

This paper investigates the extendibility of certain P_{k} sets to Diophantine quadruples, demonstrating non-extendibility for specific values of k and exploring properties of these sets.

## Contribution

It establishes non-extendibility results for P_{k} sets with k=2 and k=-3 and provides new properties of P_{k} sets related to Diophantine m-tuples.

## Key findings

- P_{k} sets with k=2 cannot be extended to Diophantine quadruples.
- P_{k} sets with k=-3 cannot be extended to Diophantine quadruples.
- The paper identifies specific properties of P_{k} sets relevant to their extendibility.

## Abstract

A set of m distinct positive integers {a_{1},...a_{m}} is called a Diophantine m-tuple if a_{i}a_{j}+n is a square for each 1\leqi<j\leqm . The aim of this study is to show that some P_{k} sets can not be extendible to a Diophantine quadruple when k=2 and k=-3 and also to give some properties about P_{k} sets.