Markoff-Rosenberger triples in arithmetic progression
We study the solutions of the Rosenberg-Markoff equation ax(2) + by(2) + cz(2) = 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. (C) 2012 Elsevier B.V. All rights reserved.
arithmetic progression markoff equation