On the Difference Between An Integer and Its Inverse Modulo n Original Research Article

Let n > 2 be an integer, and for each integer 0 < x < n with (n, x) = 1, define image by the congruence ximage ≡ 1 (mod n) and 0 < image < n. Let M(n, k) = ∑′n−1a=1 (a − image)2k. The main purpose of this paper is to study the asymptotic behaviour of M(n, k), and prove for any positive integer k that we have [formula] where φ(n) is the Euler function, d(n) is the divisor function, and the O(·) provides an absolute constant.