Date: Tue, 20 Jul 1999 21:10:07 0700
Sender: Michael & Stephanie
I am a college student but this problem is for high school math.
if you have the equation x= n^{2}  m^{2} (ie 40= 7^{2}3^{2}= 499) x must = a positive number
1) which squared numbers work as n and m
2) how does it work
3) if my teacher gave me the number for x; how could I figure out this problem
Thanks for your help
Hi
The pattern that you need to see here is the difference of squares. That is for any numbers m and n
x = m^{2}  n^{2} = (m  n )(m + n)
Notice that (m  n) + (m + n) = 2m which is even. Hence if x can be written as the difference of two squares then it must be possible to write x as the product of two numbers whose sum is even.
Numbers x that are divisible by 2 but not by 4, such as 6 or 30, cannot be written as the difference of two squares since if you write such a number x as a product, say k times j then one of k and j is even and the other is odd so k + j is odd. On the other hand if x is divisible by 4 then you can write x = 2(2s) for some number s and 2 + 2s is even. If x is odd and you write x as a product of k and j then both k and j are odd so k + j is even.
Now suppose that x is k times j and k + j is even. Let m = (k + j)/2. You should now be able to find n so that
x = (m  n )(m + n) = m^{2}  n^{2}
Thus the only integers x for which you cannot find integers m and n such that x = m^{2}  n^{2} are integers that are divisible by 2 but not by 4.
Cheers
Harley
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