Question

Prove that every odd integer is the difference of two squares

Answer 1

I can disprove it
1 is odd
the square are as follows
1,4,9,16,25
notice the difference becomes bigger
the smallest difference is 4-1=3
not 1
disproved

Answer 2

Piece of cake, start with any odd integer o and it can be written as o=2n-1 where n is an integer
add and subtract n^2 o=n^2-n^2+2n-1
rewrite this as o=n^2-(n^2-2n+1)
and factor o=n^2-(n-1)^2
and there you have it. The odd number is written as the difference of two perfect squares. In fact, what I just proved is a little bit stronger. I proved that every odd integer is the difference of two CONSECUTIVE perfect squares.