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Abc Conjecture

Statement

For any ϵ>0, there exists a constant C_ϵ such that for any three relatively prime integers a, b, c satisfying a + b = c, the inequality max( left bracketing bar a right bracketing bar , left bracketing bar b right bracketing bar , left bracketing bar c right bracketing bar )<=C_ϵ product_(p|abc)p^(1 + ϵ) holds.

Formal statement

$for all _(ϵ, ϵ>0)exists_(C_ϵ) for all _({a, b, c}, (a, b, c) element Z^3 ∧ gcd(a, b, c) = 1 ∧ a + b = c)max( left bracketing bar a right bracketing bar , left bracketing bar b right bracketing bar , left bracketing bar c right bracketing bar )<=C_ϵexp( sum_(n∣abc ∧ n element P)(1 + ϵ)log(n))$

History

formulation date | 1985 (40 years ago)
formulators | Joseph Oesterlé | David Masser
status | open

Associated equation

max( left bracketing bar a right bracketing bar , left bracketing bar b right bracketing bar , left bracketing bar c right bracketing bar )<=C_ϵ product_(p|abc)p^(1 + ϵ)

Classes

mathematical conjectures | unsolved mathematics problems

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