Difference of Squares of Two Integers

Date: 7/2/96 at 8:2:7
From: Anonymous
Subject: Difference of Squares of Integers

1.  Which positive integers can be written as the difference 
of the squares of two integers?  Explain your reasoning.

I know that the numbers that are squared cannot be equal, so I came up 
with this:  n^2 - (n-x)^2 = a positive integer.  The smallest number 
that will work is 3 (2^2 - 1^2 = 3). Continuing sequentially, the 
differences turn out to be odd numbers.  If I try non-sequential 
numbers, however, the difference can be even. (6^2 - 4^2 = 20)

I have an equation, but don't know how to make it work.  Can 
you help?

Thanks, Nancy

Date: 7/2/96 at 10:49:52
From: Doctor Anthony
Subject: Re: Difference of Squares of Integers
Any expression of the form a^2 - b^2 can be factorized as (a-b)(a+b).

It follows that if a number can be factorized in this way, then it can 
be expressed as a difference of squares.

The example you gave could be worked as follows:

20 = 10*2

     So a+b = 10
        a-b =  2
       ---------
   add 2a   = 12    so a = 6
subtract 2b = 8     so b = 4

You will note from this working that the sum of the factors must be 
even, since they will equal 2a (and the difference will equal 2b).  So 
if a number can be factorized into two even or two odd factors, then 
it can be expressed as a difference of squares.

Example.  35 = 7*5     a+b = 7
                       a-b = 5
                      ---------
                        2a = 12   and a = 6   b = 1

and it is true  35 = 36-1 = 6^2 - 1^2

Example  63 = 9*7     a+b = 9   a-b = 7  2a = 16  a=8  b=1
         and 8^2-1^2 = 63 

Example  60 = 10*6    a+b = 10  a-b = 6  2a = 16  a=8  b=2
         and 8^2-2^2 = 60

So the general rule is that provided a number can be factorized with 
two even or two odd factors, then it can be represented as a 
difference of squares.

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