 Abc conjecture

For every $ϵ>0$ there is a constant $K_{ϵ}$ such that the displayed inequality holds for coprime positive integers $A+B=C$. The radical is the product of the distinct primes dividing the argument. Applying it to a primitive Fermat triple bounds the exponent and the triple simultaneously. For $a{x}^n-b{y}^n=c\ne 0$, division by the gcd, which divides $c$, gives a uniform bound on $\max (x,y{)}^{n-2(1+ϵ)}$. Finiteness in the varying exponent requires excluding $x=y=1$ when $a-b=c$.