How many natural numbers below 1000 can be written as difference of squares of two natural numbers?

Answer 1

All but 251 of them.

Any odd number greater than 1 can be written in the form 2n + 1 where n = 1,2,3,...

then 2n+1 = (n + 1)2 - n2

Any multiple of 4 can be written as 4n where n = 1,2,3,...

then 4n = (n + 1)2 - (n - 1)2

However, unless you allow 1 = 12 - 02

1 cannot be so written as a difference of two squares

and odd multiples of 2 cannot be so written.

So that is the number 1 and 250 odd multiples of 2: 251 numbers in all.

1 cannot be so written.