Integral solutions of Markoff-Hurwitz equations
The author discusses the positive integral solutions to the symmetric equations of Hurwitz, \(x_ 1^ 2+ \dots+ x_ n^ 2= ax_ 1\cdots x_ n\). Markoff showed that in the case of \(a=n =3\), there is a fundamental solution from which a straightforward group action generates all of the positive integral solutions. Hurwitz proved that the set of positive integral solutions for a general pair \(a\) and \(n\) is either empty, or arises from a finite number of fundamental solutions. In this well-written paper, the author sharpens results of N. P. Herzberg on the set of \(a\) and \(n\) which lead to non-trivial solution sets. He also constructs, in two rather different manners (one completely elementary, the other involving the theory of quadratic number fields), sequences of \(a\) and \(n\) with unbounded number of fundamental solutions.
scientific article; zbMATH DE number 653305