We determine the number of positive integral points on $n$-dimensional affine varieties associated to arbitrary $n \times n$ generalized Cartan matrices. An application to the theory of cluster algebras and combinatorics is the resolution of the Fontaine-Plamondon conjecture, which says that there are exactly $4400$ and $26952$ positive integral friezes of type $E_7$ and $E_8$ respectively. An application to number theory refines and generalizes theorems of Mohanty, Mordell, and Schinzel to the positive integers and higher dimensions by exhibiting examples of Diophantine equations $xyz = G(x, y)$ and $xyzw = G(x, y, z)$ of every degree greater than $3$ with infinitely many positive integral solutions.