Lecture on the abc Conjecture and Some of Its Consequences

Waldschmidt, Michel

doi:10.1007/978-3-0348-0859-0_13

Cited by 7 publications

(13 citation statements)

References 59 publications

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“…This simply stated but very deep Diophantine problem has far-reaching implications in number theory [4,5].…”

Section: Introductionmentioning

confidence: 99%

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Squarefree integers and the $abc$ conjecture

Batang

2021

Preprint

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For coprime positive integers a, b, c, where a + b = c, gcd(a, b, c) = 1 and 1 ≤ a < b, the famous abc conjecture (Masser and Oesterlè, 1985) states that for ε > 0, only finitely many abc triples satisfy c > R(abc) 1+ε , where R(n) denotes the radical of n. We examine the patterns in squarefree factors of binary additive partitions of positive integers to elucidate the claim of the conjecture. With abc hit referring to any (a, b, c) triple satisfying R(abc) < c, we show an algorithm to generate hits forming infinite sequences within sets of equivalence classes of positive integers. Integer patterns in such sequences of hits are heuristically consistent with the claim of the conjecture.

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“…This simply stated but very deep Diophantine problem has far-reaching implications in number theory [4,5].…”

Section: Introductionmentioning

confidence: 99%

“…Here, we call any abc triple satisfying R(abc) < c an abc hit [4], e.g. (1, 2 3 , 3 2 ), (5, 3 3 , 2 5 ), (2, 3 10 • 109, 23 5 ) and ( 11 where Q(c) indicates how far c is from being a squarefree or, equivalently, how rich c in squareful factors [6].…”

Section: Introductionmentioning

confidence: 99%

Squarefree integers and the $abc$ conjecture

Batang

2021

Preprint

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“…16 There are also so-called explicit stronger versions of the abc conjecture, which easily imply the Wiles-Fermat theorem, see e.g. [49], and which are not dealt with in [31]. For a discussion of the relationship between [31] and solutions to the Fermat equation see the final paragraph of 2.12.…”

Section: Ivan Fesenkomentioning

confidence: 99%

“…For an entertaining presentation of aspects of the abc conjecture and related properties, see e.g. [49]. 8 In 1978 Szpiro talked about it with several mathematicians.…”

Section: Ivan Fesenkomentioning

confidence: 99%

“…This procedure allows one to estimate the value group portions of various monoids of arithmetic interest in terms of their unit group portions and underlies the proof of the inequality in the main theorem. 49 While local class field theory is used in IUT, global class field theory is not. It is crucial for IUT to use the full Galois and arithmetic fundamental groups.…”

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confidence: 99%

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Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta-functions, notes on the work of Shinichi Mochizuki

Fesenko¹

2015

European Journal of Mathematics

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ABSTRACT. These notes survey the main ideas, concepts and objects of the work by Shinichi Mochizuki on interuniversal Teichmüller theory [31], which might also be called arithmetic deformation theory, and its application to diophantine geometry. They provide an external perspective which complements the review texts [32] and [33]. Some important developments which preceded [31] are presented in the first section. Several important aspects of arithmetic deformation theory are discussed in the second section. Its main theorem gives an inequality-bound on the size of volume deformation associated to a certain log-theta-lattice. The application to several fundamental conjectures in number theory follows from a further direct computation of the right hand side of the inequality. The third section considers additional related topics, including practical hints on how to study the theory. CONTENTS

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Good elliptic curves with a specified torsion subgroup

Barrios

2023

Journal of Number Theory

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Lecture on the abc Conjecture and Some of Its Consequences

Cited by 7 publications

References 59 publications

Squarefree integers and the $abc$ conjecture

Squarefree integers and the $abc$ conjecture

Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta-functions, notes on the work of Shinichi Mochizuki

Good elliptic curves with a specified torsion subgroup

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