= Primitive Fermat bound from abc
{title2=$z^{n-7/2}\le C_{1/6}$}
For a primitive positive solution of $x^n+y^n=z^n$ with $n\ge4$, the three powers are pairwise <coprime> and their radical is at most $xyz<z^3$. The <abc conjecture> with $\epsilon=1/6$ gives the displayed inequality, hence $z\le C_{1/6}^2$ uniformly in $n$. The inequality $z^n\le2\max(x,y)^n$ then bounds $n$. Thus there are finitely many primitive triples and exponents. Homogeneity would produce infinitely many unscaled solutions from any one nonzero solution, so a primitive <arithmetic height> bound alone must not be described as finite many arbitrary multiples.
Back to article page
