On the Unicity Conjecture for Generalized Markoff Equation

Abstract

The Markoff equation x² + y² + z² = 3xyz is introduced by A.A. Markoff in 1879. A famous conjecture on the Markoff equation, made by Frobinus in 1913, states that any Markoff triples (x, y, z) with x ≤ y ≤ z is uniquely determined by its largest number z. The complete solution of this equation is still open however the partial solution is given by Barager (1996), Button (2001), Zhang (2007), Srinivasan (2009), Chen and Chen (2013). In 1957, Mordell developed a generalization to the Markoff equation of the form x² + y² + z² = Axyz + B where, A and B are positive integers. In 2015, Donald McGinn take a particular form of above equation with A = 1 and B = A and gave a partial solution to the unicity conjecture to this equation. In this paper, the partial solution to the unicity conjecture to the equation of the form x² + y² + z² =3xyz + A where A is positive integer with A ≤ 4(x² –1) is given. 

Journal of Advanced College of Engineering and Management, Vol. 3, 2017, Page : 137-145