abc Conjecture

abc Conjecture

The abc conjecture (also known as Oesterlé–Masser conjecture) is a conjecture in number theory, first proposed by Joseph Oesterlé (1988) and David Masser (1985) as an integer analogue of the Mason–Stothers theorem for polynomials. The conjecture is stated in terms of three positive integers, a, b and c (hence the name), which have no common factor and satisfy a + b = c. If d denotes the product of the distinct prime factors of abc, the conjecture essentially states that d is usually not much smaller than c. In other words: if a and b are divisible by large powers of primes, then c is usually not divisible by large powers of primes. The precise statement is given below.

The abc conjecture has already become well known for the number of interesting consequences it entails. Many famous conjectures and theorems in number theory would follow immediately from the abc conjecture. Goldfeld (1996) described the abc conjecture as "the most important unsolved problem in Diophantine analysis".

In August 2012, Shinichi Mochizuki released a series of four preprints containing a serious claim to a proof of the abc conjecture. Mochizuki calls the theory on which this proof is based "inter-universal Teichmüller theory", and it has other applications including a proof of Szpiro's conjecture and Vojta's conjecture. Experts were expected to take months to check Mochizuki's new mathematical machinery, which was developed over decades in 500 pages of preprints and several of his prior papers.