How would you go about finding prime factors of a number like $7999973$? I have trivial knowledge about divisor-searching algorithms.
The thing to notice here is that 7,999,973 is close to 8,000,000. In fact it is $8000000 - 27$. Both of these are perfect cubes. Differences of cubes always factor: $$a^3 - b^3 = (a-b)(a^2+ab+b^2)$$
Here we have $a=200, b=3$, so $a-b= 197$ is a factor.