Abc conjecture

Statement

For every $ϵ>0$, there exists a constant $C_{ϵ}$ such that for any three relatively prime integers $a,b,c$ such that:

$a+b=c$,

we have the inequality:

$max(|a|,|b|,|c|)\le C_{ϵ}{\prod }_{p|abc}{p}^{1+ϵ}$

where the indicated product is only over prime divisors of the product $abc$.

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 $ϵ$.

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.