Primitive Fermat bound from abc ($z^{n-7/2}\le C_{1/6}$)

For a primitive positive solution of $x^n+y^n=z^n$ with $n\ge 4$, the three powers are pairwise coprime and their radical is at most $xyz<z^3$. The abc conjecture with $ϵ=1/6$ gives the displayed inequality, hence $z\le C_{1/6}^2$ uniformly in $n$. The inequality $z^n\le 2\max (x,y{)}^n$ 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.