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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.
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 ...
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 ...
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. [2] The theorem states:
A proof of this conjecture, together with the more powerful geometrization conjecture, was given by Grigori Perelman in 2002 and 2003. Perelman's solution completed Richard Hamilton 's program for the solution of the geometrization conjecture, which he had developed over the course of the preceding twenty years.
Joseph Oesterlé (born 1954) is a French mathematician who, along with David Masser, formulated the abc conjecture which has been called "the most important unsolved problem in diophantine analysis". [2] [3] He is a member of Bourbaki. [4]
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.