The extension of some D(4)-pairs

TL;DR

This paper demonstrates that certain D(4)-triples can be uniquely extended to quadruples and cannot be extended further to quintuples within specific bounds, advancing understanding of D(4)-pair extensions.

Contribution

It proves the uniqueness of D(4)-triple extensions to quadruples and establishes bounds preventing extension to quintuples.

Findings

Unique extension of D(4)-triples to quadruples within bounds.

D(4)-pairs cannot be extended to quintuples under certain conditions.

Provides bounds for extension possibilities of D(4)-pairs.

Abstract

In this paper we illustrate the use of the results from [1] proving that $D(4)$-triple $\{a, b, c\}$ with $a < b < a + 57\sqrt{a}$ has a unique extension to a quadruple with a larger element. This furthermore implies that $D(4)$-pair $\{a, b\}$ cannot be extended to a quintuple if $a < b < a + 57\sqrt{a}$.