David Spain wrote:

Perhaps a logical
system would be based on only representing prime numbers. Any other
numbers could be made up of symbols for primes indicating multiplying
them to get composite numbers.

In fact there is the well-known Chinese Remainder Theorem that is based
on powers of primes. It has the unique ability to add and multiply numbers
without the use of digit carries. The drawback is that there is more than
one representation for the same number. And no one has really figured out
a good way to do division this way.

I'm not sure why you say that. You can define numbers in a unique way
using a system based on powers of their prime factors.

If you define 0 and 1 separately, then you can have a unique representation
for all integers. You put the prime factors in say increasing order, their powers
will be well defined. Once integers are uniquely defined, rational numbers
can be uniquely defined in the usual way, then real and complex numbers can
also be constructed in the usual way (and quaternions if so you wish).

Of course if the number 1 is not posited to exist independently of prime
power system, then you have to say something like 1 = 2^0. But if you accept
to put primes to the power zero in your number definition system then you
get into non unique representation problems (or you can avoid the non unique
representation problem by representing numbers by the entire infinite series
of primes with powers, powers which will all be zero safe for a finite set,
this brings in serious problems).


Alain Fournier