On Wed, 18 Sep 2013, Brüder des Schattens Söhne des Lichts wrote:
> Theorem 1: For any given odd integer that does not contain the integer 5 > as a factor, there exists an infinite number of integers (consisting only of > digit 9's, e.g., 999999.) which contain this given odd integer as a factor.
What the heck are you saying? Is the following what you're stating?
Let n be an odd integer not divisible by 5, then there are infinitely many integers of the form 10^k - 1, k in N with n dividing 10^k - 1.
Proof. Clearly the proposition holds for n = 1. Since, neither 2 nor 5 divides n, 10 and n are coprime. Thus by Euler's theorem, for all k in N, 10^(k.phi n) = 1 (mod n).