Infinite carry chains

· You may have noticed that repetends can have carries across repetend boundaries… I’ve developed methods for dealing with this, which is the subject of this paper.

First, a few quick notes on terminology…

I’ve been using the Euler Phi function, , inconsistently… and inappropriately. As I’ve now recalled, . Although not appropriate, until I can find, or someone can show me, better terminology, for this paper I’m going to redefine . The standard Phi function would then be .
I need to define a new function, . When determining the lengths of the terminating and repeating portion of the fractional part of a number, I need to determine the portion of the denominator of the fraction representation of the number that is coprime to the base of the fractional expansion I’m determining, and also the portion of the denominator that is not coprime to this base. I’ve been taking the GCD of the base and the denominator of the fraction representation of the number as the portion that is not coprime to the base of the fractional expansion being sought. As is illustrated in an example here, the GCD is not (usually not…) the same as the portion that is not coprime to the base. So far, I’ve just been lucky to pick examples where they are the same.

Definition of :

(1.1)

Thus, is the portion of not coprime with . Consequently, is the portion of coprime with .

Determination of :

1. Given the prime power factorizations of and :

That is, to determine take all primes common to both and and raise each of these to the power each is raised to in .

2. Recursive or iterative implementation of the least positive remainder GCD algorithm:

o Of course, this second method is probably more suitable for machine implementation.

Now, I’ll demonstrate the new methods for dealing with infinite carry chains of repetends:

And below is an example of an infinite carry chain…

The next one’s a little more involved…

I suspect someone’s going to be wondering how I determined all the factors of 420… so

I wrote this short paper on finding all factors of a number…

This is an interesting case…

Or… the more general case…