abc conjecture

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. It is stated in terms of three positive integers ${\displaystyle a,b}$ and ${\displaystyle c}$ (hence the name) that are relatively prime and satisfy ${\displaystyle a+b=c}$. The conjecture essentially states that the product of the distinct prime factors of ${\displaystyle abc}$ is usually not much smaller than ${\displaystyle c}$. 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".

The abc conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves, 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.

Various attempts to prove the abc conjecture have been made, but none are currently accepted by the mainstream mathematical community, and, as of 2023, the conjecture is still regarded as unproven.