Originally Posted by cjameshuff

For example, every positive integer has a unique representation as a product of integer powers of primes, and can be treated as a vector of those powers.

Primes: 2, 3, 5, 7, 11, 13...

1 = <0> = 2^0
2 = <1> = 2^1
3 = <0, 1> = 2^0 * 3^1
4 = <2> = 2^2
12 = <2, 1> = 2^2 * 3^1
15 = <0, 1, 1> = 2^0 * 3^1 * 5^1
42 = <1, 1, 0, 1> = 2^1 * 3^1 * 5^0 * 7^1

If 1 is a prime, you must specifically treat it as special and omit it from the above representation, because 1^n is always 1...there would be an infinite number of representations for each of the numbers above if 1 were considered a prime.