Problem:
How many of the integers between and , inclusive, can be expressed as the difference of the squares of two nonnegative integers?
Solution:
A difference of integer squares can be factored as , the product of two integers of the same parity. Thus, for an even integer to be a difference of integer squares, it must be a product of two even integers, and therefore must be divisible by . Conversely, any integer multiple of may be expressed as the difference of two integer squares, because . Any odd number may be expressed as the difference of two integer squares, because . In all, exactly of the integers from to are differences of integer squares.
The problems on this page are the property of the MAA's American Mathematics Competitions