Hurwitz equation

Given a length $k$ and a multiplier $m$, a Hurwitz equation for a set of integers $x_1,x_2,\mathrm{\dots },x_k$ states that

$$\sum _{i=1}^{k}x_i{}^2=m\prod _{i=1}^k{x}_i.$$

There is no solution to the equation if $m>k$. Solutions can sometimes be found for $m\le k$ by starting from a $k$-long set of 1s. The Markov equation for the Markov numbers has $k=m=3$.