ABC conjecture

The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory, first proposed by Joseph Oesterlé (1988) and David Masser (1985). It is stated in terms of three positive integers, a, b and c (hence the name) that are relatively prime 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 composed from 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".

Lucien Szpiro attempted a solution in 2007, but it was found to be incorrect. In August 2012 Shinichi Mochizuki posted his four preprints which develop a new inter-universal Teichmüller theory, with an alleged application to the proof of several famous conjectures including the abc conjecture. His papers were submitted to a mathematical journal and are being refereed, while various activities to study his theory have been run. Many mathematicians remain skeptical of his work, and it may take years for the question to be resolved due to the strangeness of his proof, and other difficulties like Mochizuki's earlier resistance to leaving Japan to explain his work to others.

Before we state the conjecture we need to introduce the notion of the radical of an integer: for a positive integer n, the radical of n, denoted rad(n), is the product of the distinct prime factors of n. For example

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