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Uniqueness theorem of generalized Markov numbers that are prime powers
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abstract
In this paper, we study positive integer solutions to a generalized form of the Markov equation, given as $x^2 + y^2 + z^2 + k(yz + zx + xy) = (3 + 3k)xyz$. This equation extends the classical Markov equation $x^2 + y^2 + z^2 = 3xyz$. 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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Cited by 1 Pith paper
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Mutation-preserving generalized cluster algebras and Laurent mutation invariants
A new stability condition for generalized cluster algebras is classified, a Markov-type Diophantine equation is solved with explicit orbit counts, and the Chen-Li conjecture on rank-3 Laurent mutation invariants is proved.
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