Is Van Petrov's hypothesis about the 'impossibility' equation true?

For this complex equation with natural variables (distinct and greater than one), there is no general algorithm to prove or disprove the existence of a solution, except by complete enumeration. Moreover, attempting to prove that no such algorithm exists, as well as attempting to prove this statement, will prove more difficult than simply performing the enumeration.

k ^ n * (a^n + b^m) / (n ^ a + c) = n^k * sin(m ^ n + b ^ a) / (m ^ a - c),
where a, b, c, n, m, k ∈ N, a, b, c, n, m, k > 1, and a ≠ b ≠ c ≠ n ≠ m ≠ k.