TITLE: Find a prime factor of $7999973$ without a calculator
QUESTION [48 upvotes]: How would you go about finding prime factors of a number like $7999973$? I have trivial knowledge about divisor-searching algorithms.

REPLY [210 votes]: 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.