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Regularity criteria and an effective bound for irregular extensions of Euler triples

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Sinisa Slijepcevic ↗, Andrej Dujella ↗

Version 1 / Nov 13, 2026 / CC BY 4.0

Abstract

An Euler triple is a Diophantine triple {a,b,c}\{a,b,c\} with a<b<c=a+b+2ab+1a<b<c=a+b+2\sqrt{ab+1}. It extends to the regular Diophantine quadruple {a,b,c,d+}\{a,b,c,d_+\}, and it is conjectured that it has no other extension d>cd>c. We prove that an irregular extension forces c<4⋅1048c<4\cdot10^{48}, against c<1098c<10^{98} in recent work of He and Togb\'e. The proof squeezes the index mm of the extension, which locates a common term of two Pell sequences, between a floor from congruences, c<8m4c<8m^4, valid for every irregular extension, and ceilings from linear forms in logarithms. The same squeeze, with other bounds, gives three regularity criteria: by the size of b/ab/a, which confines irregular extensions to 1+11538<b/a<335521+\frac1{1538}<b/a<3355^2; by pair coordinates, through the hypergeometric method; and by coprime divisors of the index combinations, which settles every Euler triple with two elements in a ratio j:kj:k with jk≤300jk\le300. As applications, the three-parameter family of Cipu, Fujita and Mignotte extends only regularly when νA≥2\nu A\ge2 and K≥643K\ge643, and when K=1K=1, A∈{2,3}A\in\{2,3\} and ν≥2\nu\ge2; the latter contains the Hoggatt--Bergum triples, for which this recovers Dujella's theorem for extensions d>cd>c.

Provenance statement

This paper was prepared with the help of Aleph, an AI research assistant for mathematics, which was used to develop, organise and write up the arguments; every statement and proof was then checked step by step in a separate verification. The key ideas and the overall approach originate from the authors, and the main results have been checked by the authors. The authors take full responsibility for the content.

Tools used

Cantab Pi
Aleph

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Version history

  1. v1 · 72 pagesSubmitted by Sinisa SlijepcevicInitial depositCurrentNov 13, 2026