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The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. [ 1 ] [ 2 ] It is stated in terms of three positive integers a , b {\displaystyle a,b} and c {\displaystyle c} (hence the name) that are relatively prime and satisfy a ...
The most striking claimed application of the theory is to provide a proof for various outstanding conjectures in number theory, in particular the abc conjecture. Mochizuki and a few other mathematicians claim that the theory indeed yields such a proof but this has so far not been accepted by the mathematical community.
The Mason–Stothers theorem, or simply Mason's theorem, is a mathematical theorem about polynomials, analogous to the abc conjecture for integers. It is named after Walter Wilson Stothers, who published it in 1981, [1] and R. C. Mason, who rediscovered it shortly thereafter.
abc conjecture The abc conjecture says that for all ε > 0, there are only finitely many coprime positive integers a, b, and c satisfying a+b=c such that the product of the distinct prime factors of abc raised to the power of 1+ε is less than c. adele Adele ring algebraic number
abc conjecture The abc conjecture of Masser and Oesterlé attempts to state as much as possible about repeated prime factors in an equation a + b = c. For example 3 + 125 = 128 but the prime powers here are exceptional. Arakelov class group The Arakelov class group is the analogue of the ideal class group or divisor class group for Arakelov ...
Conjecture Field Comments Eponym(s) Cites 1/3–2/3 conjecture: order theory: n/a: 70 abc conjecture: number theory: ⇔Granville–Langevin conjecture, Vojta's conjecture in dimension 1 ⇒Erdős–Woods conjecture, Fermat–Catalan conjecture Formulated by David Masser and Joseph Oesterlé. [1] Proof claimed in 2012 by Shinichi Mochizuki: n/a ...
It would follow from the abc conjecture that there are only finitely many Brown numbers. [9] More generally, it would also follow from the abc conjecture that ! + = has only finitely many solutions, for any given integer , [10] and that ! = has only finitely many integer solutions, for any given polynomial () of degree at least 2 with integer coefficients.
Mochizuki proved Grothendieck's conjecture on anabelian geometry in 1996. He was an invited speaker at the International Congress of Mathematicians in 1998. [13] In 2000–2008, he discovered several new theories including the theory of frobenioids, mono-anabelian geometry and the etale theta theory for line bundles over tempered covers of the Tate curve.