Difference of Two SquaresDate: 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/ |
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