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The extensibility of the Diophantine triple $\{2, b, c\}$

Nikola Adžaga, Alan Filipin, Ana Jurasić

TL;DR

This work investigates whether the Diophantine triple $\{2,b,c\}$ can be extended to an irregular Diophantine quadruple. It reduces the extension problem to a Pell-system and then to an intersection problem of binary recurrences, first handling a linear form in three logarithms to bound indices, and finally applying Laurent's two-logarithm results with Baker–Davenport reduction to force small index values. The main result shows no irregular extension for certain families of $c$ depending on $b$, and a corollary states that if $\frac{b}{2}-1$ is prime, all quadruples $\{2,b,c,d\}$ with $2<b<c<d$ are regular. The methods combine Pell-type analysis, linear forms in logarithms, and Diophantine approximation tools, contributing to the understanding of the structure of Diophantine $m$-tuples and their regularity.

Abstract

The aim of this paper is to consider the extensibility of the Diophantine triple $\{2,b,c\}$, where $2<b<c$, and to prove that such a set cannot be extended to an irregular Diophantine quadruple. We succeed in that for some families of $c$'s (depending on $b$). As corollary, for example, we prove that for $b/2-1$ prime, all Diophantine quadruples $\{2,b,c,d\}$ with $2<b<c<d$ are regular.

The extensibility of the Diophantine triple $\{2, b, c\}$

TL;DR

This work investigates whether the Diophantine triple can be extended to an irregular Diophantine quadruple. It reduces the extension problem to a Pell-system and then to an intersection problem of binary recurrences, first handling a linear form in three logarithms to bound indices, and finally applying Laurent's two-logarithm results with Baker–Davenport reduction to force small index values. The main result shows no irregular extension for certain families of depending on , and a corollary states that if is prime, all quadruples with are regular. The methods combine Pell-type analysis, linear forms in logarithms, and Diophantine approximation tools, contributing to the understanding of the structure of Diophantine -tuples and their regularity.

Abstract

The aim of this paper is to consider the extensibility of the Diophantine triple , where , and to prove that such a set cannot be extended to an irregular Diophantine quadruple. We succeed in that for some families of 's (depending on ). As corollary, for example, we prove that for prime, all Diophantine quadruples with are regular.
Paper Structure (4 sections, 8 theorems, 71 equations)

This paper contains 4 sections, 8 theorems, 71 equations.

Table of Contents

  1. INTRODUCTION
  2. The system of simultaneous Pellian equations
  3. Linear form in three logarithms
  4. Linear forms in two logarithms and the proof of Theorem \ref{['Tm1']}

Key Result

Theorem 1

The triple $\{2,b,c_\nu^\pm\}$, for $\nu\in\mathbb{N}$, cannot be extended to an irregular Diophantine quadruple $\{2,b,c_\nu^\pm,d\}$, where $d>c_\nu^\pm$.

Theorems & Definitions (16)

  • Conjecture 1
  • Definition 1
  • Definition 2
  • Theorem 1
  • Corollary 1
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 6 more