Numbers that Are Both Perfect Squares and Perfect Cubes

Date: 04/10/99 at 12:42:02
From: Ben Houge
Subject: perfect squares, perfect cubes

Hi --

I am supposed to find two integers that are both a perfect square and
a perfect cube. I know what a perfect number is, but I'm not sure 
about a perfect square or perfect cube. I have looked in my 8th grade
textbook, which defines perfect number but not perfect square or cube.
I've looked in a regular dictionary, a CD encyclopedia, and I've
looked through Dr. Math online, too. Could you define those terms for
me?

Thank you!

Date: 04/11/99 at 00:02:57
From: Doctor Bruce
Subject: Re: perfect squares, perfect cubes

I sympathize with your confusion -- the word "perfect" is being used
in two very different ways!

A perfect number is a whole number the divisors of which (including 1
but not the number itself) add up exactly to the number. That should
be the definition you found in your textbook.

But perfect squares and perfect cubes are different. A perfect square
means simply a whole number which is the square of another whole
number. Likewise, a perfect cube is a whole number which is the cube
of another whole number.

For example, 25 is a perfect square (of 5), and 27 is a perfect cube
(of 3). It is true that 2 is the square of 1.4142... (a certain
decimal number), but we could not say that 2 is a *perfect* square,
because the number 1.4142... is not a whole number.

You said you wanted to find a number which is a perfect square *and* a
perfect cube at the same time. These are called "perfect sixth
powers." Some examples of these are:

     0, 1, 64, 729, 15625, ...

Hope this helps,

- Doctor Bruce, The Math Forum
  http://mathforum.org/dr.math/