On the difference between an integer and its m-th power mod n

For any real constants λ ₁, λ ₂ ∈ (0, 1], let $n geqslant max { [tfrac{1} {{lambda _1 }}],[tfrac{1} {{lambda _2 }}]} $ , m ? 2 be integers. Suppose integers a ∈ [1, λ ₁ n] and b ∈ [1, λ ₂ n] satisfy the congruence b ≡ a ^m (mod n). The main purpose of this paper is to study the mean value of (a ? b)^2k for any fixed positive integer k and obtain some sharp asymptotic formulae.