Uniqueness of Markov numbers with a large prime-power divisor

Abstract

The Frobenius uniqueness conjecture asserts that the largest entry of a positive integral solution of a^2 + b^2 + m^2 = 3abm determines the other two entries up to order. We prove that a Markov number m > 2 is unique if it has an odd prime-power divisor q satisfying ε q > 3^{-6/41} m^{29/41}, where ε = 1 for odd m and ε = 2 for even m. For odd m, this lowers the exponent 3/4 in the size condition obtained from Button's factorization criterion to 29/41. The proof associates two primitive positive representations of fourth powers by x^2 + 3mxy + y^2 to a hypothetical failure of uniqueness. An elementary bound for these representations gives the stated threshold. We also obtain a criterion involving all coprime factorizations of the odd part of m.