Question

How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two non negative integers?

Answer 1

748.

Explanation:

1. if x²-y²=(x+y)(x-y) and the both (x+y) and (x-y) are odd/even, then 'n' can be divided by 4 or 4k+-1.

2. all the numbers 4k, 4k+-1 can be written as x²-y², exept 1 and 4.

Proof:

if n=4k, then x=m+1, y=m-1 (exception m=1, n=4);

if n=2m+1, then x=m+1, y=l (exception m=0, n=1);

3. finally, all the 4k, 4k+-1 can be imagine as x²-y²;

4. in order to define the number of the required intergers:

3, 5, 7,8,9, 11,12,13, 15,16,17..., 995,996,997, 999,1000.

1000-249-1-1-1=748.

(the sequence with a₁=6, difference =4 and the last term =998).

PS. if it is possible check the solution in other sources.