The Smallest Nontrivial Solution to xk≡1 (modn) and Related Sequences

Abstract

For two positive integers $k,n\ge 2$ we consider the equation $x^k\equiv 1\hspace{1em}\left(\mathrm{mod}n\right)$ and we let s > 1 be its smallest nontrivial integer solution. We prove that as n ranges over the positive integers, the ratio $(s^k-1)/n$ takes each positive integer value infinitely often.

MSC:

• Primary 11B83, Secondary 11B50