How do you list all integers less than 20,000 that are both perfect squares and perfect cubes?

1 Answer
Jul 1, 2015

If n is a perfect square and a perfect cube then n = k^6 for some k, giving:

0^6 = 0
1^6 = 1
2^6 = 64
3^6 = 729
4^6 = 4096
5^6 = 15625

Explanation:

If an integer is both a perfect square and a perfect cube, then it will be of the form k^6 for some k in ZZ. You will find that 5^6 < 20000 < 6^6, so the only possible integers are 0^6, 1^6, 2^6, 3^6, 4^6 and 5^6. (6^6 = 46656 is too large).