Integral Solutions of Markov-Hurwitz Equations Original Research Article

In this paper, a characterization for all pairs (a, n) with a ≥ 2(n − 1)1/2 for which the equation Ma, n: x21+ · · · + x2n = ax1 · · · xn has positive integral solutions is given. It is known that for any pair (a, n), the integral solutions of Ma, n can be expressed as orbits of a finite set of fundamental solutions under the action of a group of automorphisms. Also presented in this paper are two very different constructions that yield sequences of equations whose sets of fundamental solutions grow without bound.