Generalized Markoff equations and Chebyshev polynomials

Abstract

The Markoff equation is x2+y2+z2=3xyz, and all of the positive integer solutions of this equation occur on one tree generated from (1,1,1), called the Markoff tree. In this paper, we consider trees of solutions to x2+y2+z2=xyz+A. We say a tree satisfies the unicity condition if the maximum element of an ordered triple in the tree uniquely determines the other two. The unicity conjecture says that the Markoff tree satisfies the unicity condition. In this paper, we show that there exists a sequence of real numbers {cn} such that each tree generated from (1,cn,cn) satisfies the unicity condition, and that these trees converge to the Markoff tree. We accomplish this by recasting polynomial solutions as linear combinations of Chebyshev polynomials, showing that these polynomials are distinct, and evaluating them at certain values.