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Topic: The first new theorem Primes
Replies: 3   Last Post: Sep 19, 2013 4:11 AM

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William Elliot

Posts: 1,480
Registered: 1/8/12
Re: The first new theorem Primes
Posted: Sep 19, 2013 4:11 AM
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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).



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