So every rational number can written with primes as so
[//2, //3, //5, //7, //11, //13, ...]
1 = []
2 = [1] = 2^1
3 = [0,1] = 3^1
4 = [2] = 2^2
5 = [0,0,1] = 5^1
6 = [1,1] = 2^1 * 3^1
30 = [1,1,1] = 2^1 * 3^1 * 5^1
7! = [4,2,1,1] = 2^4 * 3^2 * 5^1 * 7^1
.25 = [-2] = 2^-1
.333.. = [0,-1] = 3^-1
etc..
The advantage of this is all multiplication of numbers are simple additions
5 * 4 = 20 === [0,0,1] + [2] = [2,0,1]
and all divisions are simple subtractions
5 / 4 = 1.25 === [0,0,1] - [2] = [-2,0,1]
My question is if there's a way to add these "basePrime" number without transforming them into their original number?
aka if
30! = [26, 14, 7, 4, 2, 2, 1, 1, 1, 1]
what is 30! + 1 ?
the addition seems extremely random
> 1 + 2 = 3
[] + [1] = [0,1]
> 5 + 3 = 8
[0,0,1] + [0,1] = [3]
5040 + 4 = 5044
[4,2,1,1] + [2] = [2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1]
No, there is no known pattern. Only a rather obvious one: let $v_p(n)$ be the multiplicity of $p$ in $n$, that is, $v_p(n)=\max\{r\in\Bbb Z:p^r\mid n\}$. Then $$v_p(n+m)\begin{cases}=\min\{v_p(n),v_p(m)\}\text{ if }v_p(n)\neq v_p(m)\\ \ge v_p(n)\text{ otherwise}\end{cases}$$
Goldbach conjecture, twin primes conjecture, and surely more like those, are still conjectures. And probably they would be theorems if there were a pattern like the one you are looking for.
If you found a pattern, you should reegister your achievement and publish it. You would become rich and famous. In the $30$th century, the humankind would remember you as a great mathematician.
Maybe, even credit cards wouldn't be safe anymore... So hire some bodyguards.