The Diophantine equation α( _m ^x) + β( _n ^y) = γ

The number of solutions of the Diophantine equation ( _m ^x) + β( _n ^y) = γ with α, β, γ ∈ ℚ and m, n ∈ ℕ, m ≥ n in rational integers (x, y) is investigated. We apply the general BILU-TICHY-criterion [5] for polynomial Diophantine equations f(x) = g(y) in order to obtain ineffective finiteness of solutions (x, y) in the case m, n ≥ 3. Simplicity and two-interval-monotonicity of the local extrema of ( _m ^x) also guarantee finiteness in the case min(m, n) = 2.