Solution. We will use the well-known fact that .
If the base-2 form of the number n is , where the digits are all 0 or 1, then
Now we will show that there exists an integer r which is relatively prime to m, and an infinite sequence of positive integers such that
Let m=2tu where u is odd, and consider an arbitrary positive integer it for which (u) divides i-1. By the Euler-Fermat theorem,