Any positive integer N can be written as $ p_1^{c_1} p_2^{c_2} p_3^{c_3} ... p_n^{c_n} $, where p1, p2, p3... are the prime factors of N, and c1, c2, c3... are there respective powers. This is a unique representation of N.
If we treat p1, p2, p3 etc as the independent axes, c1 , c2, c3... can be treated as the coordinates of number N on these axes. For e.g., number 18 can be written as $ 2^13^2 $, so the coordinates of 18 are 1 and 2 along the prime axes 2 and 3 respectively.
Since this is a very simple scheme, mathematicians must have thought about it already. My question is, is there a good result that has been proven by representing integers in such a way? Is this line of thinking worth pursuing?