$D(4)$-triples with two largest elements in common

TL;DR

This paper investigates properties of $D(4)$-quadruples, proving specific cases that support conjectures about their structure, including that certain triples cannot both be $D(4)$-triples, advancing understanding of these special number sets.

Contribution

It introduces two new conjectures on $D(4)$-quadruples and proves that two specific $D(4)$-triples sharing two largest elements cannot both exist.

Findings

Proved that $ ext{set}\{a,b,c\\}$ and $ ext{set}\{a+1,b,c\\}$ cannot both be $D(4)$-triples.

Supported the validity of two new conjectures on $D(4)$-quadruples.

Provided partial results towards classifying $D(4)$-quadruples.

Abstract

In this paper we consider two new conjectures concerning $D(4)$-quadruples and prove some special cases which support their validity. The main result is a proof that $\{a,b,c\}$ and $\{a+1,b,c\}$ cannot both be $D(4)$-triples.