Orderings of Generalized k-Markov Numbers

A $k$-Markov number is a positive integer that appears in a positive integral solution to the Diophantine equation $x^2+y^2+z^2+k(xy+xz+yz)=(3+3k)xyz$. This equation was introduced by Gyoda and Matsushita. When $k=0$, this definition recovers that of ordinary Markov numbers. The set of $k$-Markov numbers can be indexed by pairs of coprime positive integers. There is a consistent way to label non-coprime pairs with positive integers as well, yielding a larger set of ``generalized $k$-Markov numbers.‘’ In this paper, we classify lines along which the generalized $k$-Markov numbers grow monotonically, extending work in the ordinary case by Lee-Li-Rabideau-Schiffler and by the second author. We find that, as $k$ grows, the $k$-Markov numbers are more likely to be monotonic along a random line. This gives evidence that a $k$-version of Frobenius’ uniqueness conjecture, which has been proposed by Gyoda and Maruyama, could be true.