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 .
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.