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

Marija Bliznac Trebješanin, Pavao Radić

TL;DR

This work analyzes extensions of $D(4)$-pairs $\{a,ka\}$ with $k \in \{7,8,10,11,12,13\}$ to $D(4)$-triples and quadruples. The authors show any resulting quadruple must be regular (i.e., $d \in \{d_{+},d_{-}\}$) by constructing the third element $c$ from Pell-type equations to form $c_{\nu}^{\pm}$, then proving that all triples extend uniquely to regular quadruples using a system of Pell equations, linear forms in three logarithms, and reduction methods (Baker–type and Dujella–Petho). They develop detailed bounds on Pell indices and intersections of recurrences, enabling a finite verification that excludes non-regular possibilities. The results support the conjecture that extensions of $D(4)$-triples to quadruples are unique and regular for these $k$-families, with the methods potentially applicable to related Diophantine m-tuple problems.

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.

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

TL;DR

This work analyzes extensions of -pairs with to -triples and quadruples. The authors show any resulting quadruple must be regular (i.e., ) by constructing the third element from Pell-type equations to form , then proving that all triples extend uniquely to regular quadruples using a system of Pell equations, linear forms in three logarithms, and reduction methods (Baker–type and Dujella–Petho). They develop detailed bounds on Pell indices and intersections of recurrences, enabling a finite verification that excludes non-regular possibilities. The results support the conjecture that extensions of -triples to quadruples are unique and regular for these -families, with the methods potentially applicable to related Diophantine m-tuple problems.

Abstract

We study the extensibility of -pairs , where and . Firstly, we show that it can be extended to a -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 -quadruple.

Paper Structure

This paper contains 9 sections, 20 theorems, 81 equations, 1 table.

Table of Contents

  1. Introduction
  2. Extensions of pairs to triples
  3. Extensions of triples and linear forms in three logarithms
  4. System of simultaneous Pellian equations
  5. A linear form in three logarithms
  6. Lower bounds for $m$ and $l$
  7. Proof of Theorem \ref{['main result']}
  8. Proof of Theorem \ref{['main result']} for $c=c_1^\pm, c_2^\pm$
  9. Proof of Theorem \ref{['main result']} with $c=c_3^\pm$

Key Result

Theorem 1.2

Let $k$ be a positive integer such that $k\in \{7, 8, 10,$$11,12,13\}.$ If $\{a, b, c, d\}$ is a $D(4)$-quadruple with $b=ka$, then it is regular. In other words, we have $d=d_{\pm}.$

Theorems & Definitions (31)

  • Definition 1.1
  • Conjecture 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 21 more