Why do we only check up to the square root of a prime number to determine if it is prime? Can't we use cube root?

Question

If a number n can be written as axb and m=sqrt(n). Here n=m*m. We say we only need to check upto m because min(a,b)<=m. So cant we take cube roots? Suppose we take n=21, then n=1x3x7. But Cube root is 2. Why does this method fail?

Answer 1

Consider n = 143 = 11 * 13. The cube root of 143 is between 5 and 6. If you only test divisibility by the primes up to 6, you will not find either of the two factors of n and will mistakenly conclude that 143 is prime.