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For every $ϵ>0$, the abc conjecture asserts that there is a constant $K_{ϵ}>0$ such that all coprime positive integers $A+B=C$ satisfyThe radical of an integer is the product of its distinct prime factors. The condition $\gcd (A,B)=1$ makes the three entries pairwise coprime. Repeated prime powers enlarge a number without enlarging its radical in the same way, so the conjecture constrains how often addition can relate unusually concentrated prime factorizations.

The example $(A,B,C)=(1,8,9)$ has $\mathrm{rad}(ABC)=6<9$ and is an abc-hit. Such examples are compatible with the conjecture: its bound includes both the constant $K_{ϵ}$ and the exponent $1+ϵ$. Conditional applications include the primitive Fermat bound from abc and the Catalan power bound from abc.

More generally, fix positive integers $a,b$, a nonzero integer $c$, and positive integers $x,y,n$ satisfying $a{x}^n-b{y}^n=c$. Put $g=\gcd (a{x}^n,b{y}^n)$, which divides $|c|$. Applying the conjecture to the normalized positive triple formed from $a{x}^n$, $b{y}^n$ and $|c|$ gives, for $M=\max (x,y)$,Indeed, the largest normalized entry is at least a fixed multiple of $M^n$, while the radical is at most $\mathrm{rad}(ab|c|)xy\le \mathrm{rad}(ab|c|)M^2$. Taking $ϵ=1/4$ makes the exponent at least $1/2$ for $n\ge 3$, bounding $M$ uniformly; if $M\ge 2$, it also bounds $n$. The separate case $x=y=1$ can satisfy the equation for every $n$ when $a-b=c$.