The Life of a Mathematician

Okay, I haven't made a mathematical post in a while, and I want to explore an idea I just had. The positive integers form a group under multiplication. Call it G. G is infinitely generated, and the most natural basis set for G would be P, the set of prime numbers. Come to think of it, that's the only possible basis set. Any given prime would have to be in it, since a prime can't be written as a product of other integers, and once you've got all the primes, putting any other integer into the set would violate the uniqueness criterion. Okay, so we've got G which is generated by P. Let's consider endomorphisms of G. If f: G -> G is an endomorphism, then f is completely determined by where it sends elements of P. Here's the thought I had earlier: Suppose f(2) = 3, f(3) = 2, f(p) = p for p in P\{2,3}. I thought this might shed some light on the Collatz problem since it deals so much with 2's and 3's. Anyway, let's look at a table:

1 - 1
2 - 3
3 - 2
4 - 9
5 - 5
6 - 6
7 - 7
8 - 27
9 - 4
10 - 15
11 - 11
12 - 18
13 - 13
14 - 21
15 - 10
16 - 81
17 - 17
18 - 12
19 - 19
20 - 45
21 - 14
22 - 33
23 - 23
24 - 54
25 - 25

Okay, I can see one thing already. f(x) = y <=> f(y) = x. So f is its own inverse. Interesting. You could also set up something more complicated, I suppose. Like, put all the primes into pairs and swap them. Okay, I've run out of ideas for the moment. Brainstorm finished.