##### Wiktionary

**abc conjecture**

n. (context math English) A conjecture in number theory, stated in terms of three positive integers, ''a'', ''b'' and ''c'', 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 composed from large powers of primes, then ''c'' is usually not divisible by large powers of primes.

##### Wikipedia

**Abc conjecture**

The ** abc conjecture** (also known as the

**Oesterlé–Masser conjecture**) is a conjecture in number theory, first proposed by and . 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. 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 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.