The product of distinct prime factors of a,b,c, where c is a+b, is rarely much less than c
abc conjecture
Field
Number theory
Conjectured by
Joseph Oesterlé
David Masser
Conjectured in
1985
Equivalent to
Modified Szpiro conjecture
Consequences
Beal conjecture
Erdős–Ulam problem
Faltings's theorem
Fermat's Last Theorem
Fermat–Catalan conjecture
Roth's theorem
Tijdeman's theorem
Mathematician Joseph OesterléMathematician David Masser
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 and (hence the name) that are relatively prime and satisfy . The conjecture essentially states that the product of the distinct prime factors of is usually not much smaller than . A number of famous conjectures and theorems in number theory would follow immediately from the abc conjecture or its versions. Mathematician Dorian Goldfeld described the abc conjecture as "The most important unsolved problem in Diophantine analysis".[3]
The abc conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves,[4] which involves more geometric structures in its statement than the abc conjecture. The abc conjecture was shown to be equivalent to the modified Szpiro's conjecture.[1]
Various attempts to prove the abc conjecture have been made, but none have gained broad acceptance. Shinichi Mochizuki claimed to have a proof in 2012, but the conjecture is still regarded as unproven by the mainstream mathematical community.[5][6][7]
^ abOesterlé 1988.
^Masser 1985.
^Goldfeld 1996.
^Fesenko, Ivan (September 2015). "Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta-functions, notes on the work of Shinichi Mochizuki". European Journal of Mathematics. 1 (3): 405–440. doi:10.1007/s40879-015-0066-0.
^
Ball, Peter (10 September 2012). "Proof claimed for deep connection between primes". Nature. doi:10.1038/nature.2012.11378. Retrieved 19 March 2018.
^Castelvecchi, Davide (9 April 2020). "Mathematical proof that rocked number theory will be published". Nature. 580 (7802): 177. Bibcode:2020Natur.580..177C. doi:10.1038/d41586-020-00998-2. PMID 32246118. S2CID 214786566.
^Further comment by P. Scholze at Not Even Wrong math.columbia.edu[self-published source?]
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