A cluster-mutation argument claims the Diophantine equation x1^2+x2^4+x3^4+2x1x2^2+2x1x3^2 = k x1 x2^2 x3^2 has positive integer solutions only for k=7, alongside a classification of sign-equivalent exchange matrices.
Aigner Markov's theorem and 100 years of the uniqueness conjecture: a mathematical journal from irrational numbers to perfect matchings Springer 2013
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A cluster theory approach from mutation invariants to Diophantine equations
A cluster-mutation argument claims the Diophantine equation x1^2+x2^4+x3^4+2x1x2^2+2x1x3^2 = k x1 x2^2 x3^2 has positive integer solutions only for k=7, alongside a classification of sign-equivalent exchange matrices.