Markoff-Rosenberger triples in arithmetic progression

Abstract

We study the solutions of the Rosenberg–Markoff equation ax2 + by2 + cz2 = dxyz (a generalization of the well–known Markoff equation). We specifically focus on looking for solutions in arithmetic progression that lie in the ring of integers of a number field. With the help of previous work by Alvanos and Poulakis, we give a complete decision algorithm, which allows us to prove finiteness results concerning these particular solutions. Finally, some extensive computations are presented regarding two particular cases: the generalized Markoff equation x 2 +y 2 +z 2 = dxyz over quadratic fields and the classic Markoff equation x 2 + y 2 + z 2 = 3xyz over an arbitrary number field.