Counting Minimal Triples for a Generalized Markoff Equation

Abstract

If the generalized Markoff equation a 2 + b 2 + c 2 = 3 abc + m has a solution triple, then it has infinitely many solutions. For a positive integer m > 1, we show that all positive solution triples are generated by a finite set of triples that we call minimal triples. We exhibit a correspondence between the set of minimal triples with the first or second element equal to a, and the set of fundamental solutions of m − a 2 by the form x 2 − 3 axy + y 2 . This gives us a formula for the number of minimal triples in terms of fundamental solutions, and thus a way to calculate minimal triples using composition and reduction of binary quadratic forms, for which there are efficient algorithms. Additionally, using the above correspondence we also give a criterion for the existence of minimal triples of the form ( 1 , b , c ) , and present a formula for the number of such minimal triples.