An equivalent condition for the Markov triples and the Diophantine equation $a^2+b^2+c^2=abcf(a,b,c)$

Abstract

We propose an equivalent condition for the Markov triples, which was mentioned by H. Rademacher essentially. As an application, we study the solvability of the Diophantine equation $a^2+b^2+c^2=abcf(a,b,c)$.