Difference of Two Squares

Date: 08/17/97 at 06:56:22
From: Zeshan
Subject: Algebra

Prove that every cube can be expressed as the difference of two 
squares.

Date: 08/22/97 at 10:07:32
From: Doctor Rob
Subject: Re: Algebra

Hint:

a*a^2 = a^3 = x^2 - y^2 = (x-y)*(x+y)

Notice that x - y and x + y are either both odd or both even.  Notice
that a and a^2 are either both odd or both even.

Alternate hint:

Every odd number and every multiple of 4 can be written as a 
differenceof squares, and a^3 is always either one or the other.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 08/23/97 at 12:08:46
From: Zeshan Ghory
Subject: Re: Algebra

I was working with your hints (and not getting too far), but then I 
suddenly realised a simple solution -
        
     r^3 = sum of  [(r^3) between r=1 and n] - sum of [(r-1)^3 
           between same limits]

  => r^3 = (1/4)(n^2)[(n+1)^2] - (1/4)[(n-1)^2](n^2)        
           {Using standard formulae for r^3}

  => r^3 = [(1/2)(n)(n+1)]^2 - [(1/2)(n-1)(n)]^2 

Which is, of course, a difference of two squares

So   a^3 = x^2 - y^2

where      

     x^2 = (1/2)[(a^2)+a]  and y = (1/2)[(a^2)-a]

Thanks.

Zeshan

Date: 08/29/97 at 10:42:19
From: Doctor Rob
Subject: Re: Algebra

You would get the same solution if you set x - y = a, x + y = a^2,
and solved for x and y, which is what I intended for you to find 
using the first hint.

If a is odd, then x - y = 1 and x + y = a^3 will give a solution, 
and if a is even, then x - y = 2 and x + y = a^3/2 will give a 
solution, which is what I intended for you to find using the alternate 
hint.

Your solution is very clever.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/