Has it been proven that all rational numbers repeat ....

[swampwiz]

in every radixial representation, except of course for those cases in which the numerator is a factor of some natural-number power of the radix?

For the radixial system we know (i.e., because we are bilateral and have arms that have 5 fingers), this would mean that any possible natural number that does not have 2 or 5 as factor (which if it did have such a factor would mean that it is a factor of some number 10ⁿ) must be a factor of some number 9999 ..., and of course mean something similar for a radixial number of any radix?

(I am probably using some improper terminology, including the term radixial, but I think folks understand what I mean here - i.e., as a generic term for a number that is in decimal, hexadecimal, binary, etc.)

 

[haruspex]

in every radixial representation, except of course for those cases in which the numerator is a factor of some natural-number power of the radix?

For the radixial system we know (i.e., because we are bilateral and have arms that have 5 fingers), this would mean that any possible natural number that does not have 2 or 5 as factor (which if it did have such a factor would mean that it is a factor of some number 10ⁿ) must be a factor of some number 9999 ..., and of course mean something similar for a radixial number of any radix?

(I am probably using some improper terminology, including the term radixial, but I think folks understand what I mean here - i.e., as a generic term for a number that is in decimal, hexadecimal, binary, etc.)

In the process of division of one integer m by a coprime integer n, using some base b, at each step there is a remainder less than n. So in n steps or fewer, this remainder must recur. All of the resulting digits will therefore recur from that point, ad infinitum.