There exist number systems where addition can be done without using carries which makes the addition of two small numbers as fast as the addition of two large numbers when using a computer. The redundant binary signed digit system has this property.
Does there exist a number system where the same is true for multiplications (multiplication of small numbers is as fast as multiplication of large numbers on a computer)?
My first thought is something like canonicalized prime factorizations: $$ N = p_1^{k_1} p_2^{k_2}\cdots p_n^{k_n} $$
Where $p_n$ is the nth prime. By the fundamental theorem of arithmetic, there is always a unique representation in this form, though $n$ may become arbitrarily large. By the prime number theorem, $n$ will tend to grow logarithmically in proportion to the largest representable value, which is similar to the behavior of other number systems.
You would have $n$ separate RBR integers, one for each prime exponent, and each with enough bit width to represent a satisfactory range of values (again, this grows logarithmically). Multiplication is then done by addition of corresponding exponents.
The downside, of course, is that you need a lookup table of all the primes up to $p_n$ in order to convert this back to a more conventional representation. As a time-space tradeoff, you could generate such a lookup table on-demand rather than caching it indefinitely.
Unfortunately, this representation is ill-suited to most operations other than multiplying numbers, so you will likely find yourself doing these expensive conversion operations a lot. I'm not sure it's of much practical use.
