Numeric Systems...

Why a numeric system based on a polynomial?
Why not on primes, direct!
...17,13,11,7,5,3,2,1

Any integer prime decomposition is unique.

3== ...000100
15==...0001100
21== ...00010100
0== ...0000000000
1== ...0000000001 (convention: Last digit is always zero, except for the unit quantity. Alternative convention: Last digit is always one, except for the zero quantity. Convention in use is easy to spot, just by looking)

Some recursiviness is needed:
4== ...00000[...00010]0
and so on.