# Abc conjecture

From Number

## Contents

## Statement

For every , there exists a constant such that for any three relatively prime integers such that:

,

we have the inequality:

where the indicated product is only over *prime* divisors of the product .

## Related facts

### Analogous facts over other rings

- Mason-Stothers theorem states that the analogous statement holds over polynomial rings over fields with absolute value replaced by degree -- in fact, we do not even need the .

### Weaker facts and conjectures

- Logarithmic lower bound on number of non-Wieferich primes
- The abc conjecture implies Fermat's last theorem for sufficiently large primes. Fermat's last theorem was proved by Wiles in 1994, though the abc conjecture is still open.