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

1 Answer
George C.
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).

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