Question

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

Answer 1

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).