If b is an integer greater than 1, then a b-nary representation of a non-negative real number r is an expression of the form
           _∞
     r = ∑ a_ib^{-i
          i=0}
where a₀ is a non-negative integer, and the a_i are integers satisfying 0 ≤ a_i < b for i = 1,2,3, ...
     r = a₀.a₁a₂a₃ ...

An underscored numeral will denote that the numeral is repeated indefinitely. If b=10 for example, then
     0.45783 denotes 0.45783783783783....

No matter what the base b, an irrational number has only one b-nary representation.

Looking at the table below we see that (depending on the base) a rational number can have one or two b-nary representations.

---------------+-----------------------+-----------------------+
          Base |           Ten         |          Three        |
---------------+------------+----------+------------+----------+
Representation |  fraction  |  b-nary  |  fraction  |  b-nary  |                     
---------------+------------+----------+------------+----------+
               |     1/3    |   0.3    |    1/10    |  0.10    |
               |            |          |            |   or     |
               |            |          |            |  0.02    |
---------------+------------+----------+------------+----------+
               |     1/5    |   0.20   |    1/12    |  0.0121  |
               |            |    or    |            |          |         
               |            |   0.19   |            |          |       
---------------+------------+----------+------------+----------+  

If p, q, and b are integers with p ≥ 0, q > 0, and b > 1, then what is the relation between p, q, and b such that the rational number p/q has only one b-nary representation?

Basically what we want to find is if the fraction can be represented as a terminating decimal, as that can be expressed in either its terminating form, or lower the terminal digit by 1 and continue with an infinite repetition of the digit b-1.

It follows that what we want to find is that if, when p/q is reduced to its simplest form, the prime factors of the reduced q are all to be found within the prime factorization of b.  For example 3/75 reduces to 1/25, and 25 factors into 5x5. Each of these prime factors (just 5) is a factor of the base 10, so 3/75 = 1/25 can be expressed as the terminating decimal 0.04, and, like all fractions as the repeating 0.0399999....

The trick is to find the most concise way of saying that the reduced q has each of its prime factors represented by a prime factor of b.

The reduced q is just q/gcd(p,q) and it must divide evenly into some b^k where k is a positive integer.

So q/gcd(p,q) mod b^k = 0 for sufficiently large integer k will allow to b-nary representations of p/q, but non-existence of such a k will allow only one b-nary representation.

Posted by Charlie on 2009-04-27 14:59:07