Uniqueness theorem of generalized Markov numbers that are prime powers
Abstract
In this paper, we study positive integer solutions to a generalized form of the Markov equation, given as x2 + y2 + z2 + k(yz + zx + xy) = (3 + 3k)xyz. This equation extends the classical Markov equation x2 + y2 + z2 = 3xyz. We generalize the concept of Cohn triples for the classical Markov equation to the generalized Markov equations. Using this, we provide a generalization of the uniqueness theorem of Markov numbers that are prime powers.
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