Is there ever a case where it’s useful to view the natural numbers as points in an infinite dimensional vector space, the bases of which are the prime numbers and where each unit increment along any basis corresponds to that prime raised to the next higher power.

To illustrate, in this space, the number 90 (which equals 2[sup]**1**[/sup] X 3[sup]**2**[/sup] X 5[sup]**1**[/sup]) would be represented by (1, 2, 1, 0, 0, . . . ).

Likewise, 231 (which equals 2[sup]**0**[/sup]X 3[sup]**1**[/sup] X 5[sup]**0**[/sup] X 7[sup]**1**[/sup] X 11[sup]**1**[/sup]) would be represented by (0, 1, 0, 1, 1, 0, 0, . . . )

It seems like a pretty obvious thing to do, but I am too stupid to figure out whether there’s any utility in doing so.

Thanks!