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abc conjecture
The product of distinct prime factors of a,b,c, where c is a+b, is rarely much less than c / From Wikipedia, the free encyclopedia
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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 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]
Field | Number theory |
---|---|
Conjectured by | |
Conjectured in | 1985 |
Equivalent to | Modified Szpiro conjecture |
Consequences |
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Oesterle_Joseph.jpg/640px-Oesterle_Joseph.jpg)
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/e/ea/David_Masser.jpg/320px-David_Masser.jpg)
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]