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

TL;DR

This paper establishes an equivalent condition for Markov triples and applies it to analyze the solvability of a specific Diophantine equation involving these triples.

Contribution

It introduces a new equivalent condition for Markov triples and uses it to study the solvability of a related Diophantine equation.

Findings

Derived an equivalent condition for Markov triples

Analyzed the solvability of the Diophantine equation $a^2+b^2+c^2=abcf(a,b,c)$

Connected the condition to the solvability criteria of the equation

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 $a^2+b^2+c^2=abcf(a,b,c)$.