Extensions of $D(4)$-pairs $\{a, ka\}$ with $k\in \{7,8,10,11,12,13\}$

TL;DR

This paper investigates the extension properties of specific $D(4)$-pairs with multiples of a, demonstrating their unique extension to $D(4)$-quadruples and characterizing the structure of such extensions.

Contribution

It establishes the conditions under which certain $D(4)$-pairs can be extended to triples and proves the uniqueness of their extension to quadruples.

Findings

$D(4)$-pairs with $b=ka$ can be extended to triples depending on a.

Such triples have a unique extension to quadruples.

The structure of these extensions depends on a specific family of positive integers.

Abstract

We study the extensibility of $D(4)$-pairs $\{a,b\}$, where $b = ka$ and $k \in \{7,8,10,11,12,13\}$. Firstly, we show that it can be extended to a $D(4)$-triple with an element c, which is a member of a family of positive integers depending on a. Then, we prove that such a triple has a unique extension to a $D(4)$-quadruple.