Catalan power bound from abc
= Catalan power bound from abc
{c}
{title2=$x^p-y^q=1,\quad p,q\ge3\Longrightarrow x^p\le C_{1/6}^{9/2}$}
Put $M=x^p$ for positive bases $x,y\ge2$. The <coprime> triple $(y^q,1,x^p)$ has radical at most $xy<M^{2/3}$. The <abc conjecture> with $\epsilon=1/6$ gives $M\le C_{1/6}M^{7/9}$, proving the bound. Both bases and both exponents are then bounded, since $2^p\le M$ and $2^q\le M-1$. Unlike a homogeneous Fermat equation, no common-scaling ambiguity occurs.