abc conjecture

The abc conjecture predicts that for every ε > 0, only finitely many coprime positive integers a, b, c with a + b = c satisfy c > rad(abc)^(1+ε), where rad is the product of distinct prime factors.

Radical of an integer: The product of the distinct prime factors of an integer. The radical compresses the prime-factor information used to bound c.
Joseph Oesterlé: A French number theorist who formulated the abc conjecture independently of David Masser. Oesterlé developed the conjecture's formulation and its connections to arithmetic geometry.
Fermat–Catalan conjecture: The conjecture that x^m + y^n = z^k has only finitely many pairwise coprime positive-integer solutions when 1/m + 1/n + 1/k < 1. The abc conjecture implies the finiteness predicted for these generalized Fermat equations.
Shinichi Mochizuki: A Japanese mathematician who developed inter-universal Teichmüller theory and claimed a proof of the abc conjecture. Mochizuki's claimed proof remains the central disputed resolution attempt.
Coprime integers: Integers whose greatest common divisor is 1. The conjecture concerns pairwise coprime a, b, and c.
David Masser: A British number theorist known for work on Diophantine equations and the abc conjecture. Masser independently proposed the conjecture and helped bring it into number theory.
Wieferich prime: A prime p satisfying 2^(p−1) ≡ 1 modulo p². The conjecture gives strong bounds on exceptional primes of this type.
Inter-universal Teichmüller theory: A mathematical framework developed by Shinichi Mochizuki involving arithmetic geometry and multiple mathematical structures. Mochizuki's claimed derivation of abc rests on this extensive framework.
Prime factorization: The representation of an integer as a product of primes, unique up to factor order. The radical is obtained by retaining each prime factor only once.
Szpiro's conjecture: A conjecture bounding the discriminant of an elliptic curve in terms of its conductor. Oesterlé connected a reformulation of Szpiro's conjecture to abc.

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