Abc conjecture

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Consequences

Conditional proof Roth's theorem Diophantine approximation Algebraic numbers Mordell conjecture

29 related topics

Claimed proofs

Lucien Szpiro Inter-universal Teichmüller theory Peter Scholze Jakob Stix Kyoto

9 related topics

Formulations

Radical of an integer Coprime Real number Little o notation Prime numbers

5 related topics

Theoretical results

Bounded above Exponential Constants Effectively computable

4 related topics

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Conjectured by: Joseph Oesterlé · David Masser
Conjectured in: 1985
Consequences: Beal conjecture · Erdős–Ulam problem · Faltings' theorem · Fermat's Last Theorem · Fermat–Catalan conjecture
Equivalent to: Modified Szpiro conjecture
Field: Number theory

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Overview

Formulations

Examples of triples with small radical

Fermat's little theorem
Lando & Zvonkin 2004 Abc conjecture

Consequences

Conditional proof
Roth's theorem
Diophantine approximation
Algebraic numbers Algebraic number
Mordell conjecture Faltings' theorem
Gerd Faltings
Vojta's conjecture
Erdős–Woods conjecture Erdős–Woods number
Wieferich primes Wieferich prime
Marshall Hall's conjecture
Fermat's Last Theorem
A famously difficult proof by Andrew Wiles Wiles's proof of Fermat's Last Theorem
Lim sup Limit superior and limit inferior
Fermat–Catalan conjecture
L-function Dirichlet L-function
Legendre symbol
Siegel zero
Number fields Number field
Polynomial
Perfect powers
Integers
Simple zeros Simple zero
Tijdeman's theorem
The Diophantine equation n! + A = k2 Brocard's problem
Beal conjecture
Lang's conjecture Néron–Tate height
Height Height function
Erdős–Ulam problem
Siegel's theorem about integral points on algebraic curves Siegel's theorem on integral points

Theoretical results

Bounded above Upper bound
Exponential Exponential function
Constants Constant (mathematics)
Effectively computable

Computational results

Refined forms, generalizations and related statements

Claimed proofs

Lucien Szpiro
Inter-universal Teichmüller theory
Peter Scholze
Jakob Stix
Kyoto Kyoto University
The Kyoto research institute Research Institute for Mathematical Sciences
Kiran Kedlaya
Edward Frenkel
Nature Nature (journal)

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Map overview Semantic statistics

Nodes76

Edges75

Triples113

Avg. degree1.97

Density0.026316

Components1

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Abc conjecture

Top relations

related to Consequences · 40

Abc conjecture → All, An, Andrew Wiles, As, Ax, Aym, Bxn, By, Catalan, Cz, Diophantine, Dąbrowski, Erdős, Euclidean, Fermat's Last Theorem, Gerd Faltings, Granville, However, Lang's, Langevin

related to External links · 24

Abc conjecture → ABC, ABC's, ABC-Conjecture, Abderrahmane Nitaj's ABC, Barry MazurPhilosophy, Brian Hayes, Conjecture, Conjecture Numberphile, Distributed, Easy, ElkiesQuestions, Eric, Home, IUT, MathOverflowABC Conjecture Polymath, MathWorld, Mochizuki, Mochizuki's, Noam, Number Archived

related to Claimed proofs · 10

Abc conjecture → Although, He, IUTT, Lucien Szpiro, Shinichi Mochizuki, Since August, Szpiro's, Teichmüller, The, This

related to Computational results · 10

Abc conjecture → ABC, Although, As, Dutch Kennislink, Home, In, Leiden University, Mathematics Department, May, Netherlands

Consequences · 7

Abc conjecture → Beal conjecture, Erdős–Ulam problem, Faltings' theorem, Fermat's Last Theorem, Fermat–Catalan conjecture, Roth's theorem, Tijdeman's theorem

related to Theoretical results · 7

Abc conjecture → Bounds, In, K1, K2, K3, Specifically, The

related to Formulations · 5

Abc conjecture → Before, For, If, Specifically, The

related to Refined forms, generalizations and related statements · 4

Abc conjecture → Baker, Mason, Stothers, The

Conjectured by · 2

Abc conjecture → David Masser, Joseph Oesterlé

Conjectured in · 1

Abc conjecture → 1985

Important terminology Word statistics

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Important terminology

conjecture abc integers number proof displaystyle rad triples theorem theory prime many positive mochizuki claimed equivalent szpiro radical also fermat's

Entity relationships Subject–Predicate–Object triples

Subject	Predicate	Object	Confidence	Src
Abc conjecture	Conjectured by	Joseph Oesterlé	1.00	infobox
Abc conjecture	Conjectured by	David Masser	1.00	infobox
Abc conjecture	Conjectured in	1985	1.00	infobox
Abc conjecture	Consequences	Beal conjecture	1.00	infobox
Abc conjecture	Consequences	Erdős–Ulam problem	1.00	infobox
Abc conjecture	Consequences	Faltings' theorem	1.00	infobox
Abc conjecture	Consequences	Fermat's Last Theorem	1.00	infobox
Abc conjecture	Consequences	Fermat–Catalan conjecture	1.00	infobox
Abc conjecture	Consequences	Roth's theorem	1.00	infobox
Abc conjecture	Consequences	Tijdeman's theorem	1.00	infobox
Abc conjecture	Equivalent to	Modified Szpiro conjecture	1.00	infobox
Abc conjecture	Field	Number theory	1.00	infobox
Abc conjecture	is a	integer analogue of the Mason	0.90	text

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