prove that the cube of any integercan be written as the difference of two squares

Question

college math 4305 number theory

Answer 1

Consider the cube of an odd number. The difference between the squares of two consecutive integers is odd and can be expressed as 2x+1 where the squares are x and x+1. If we call the cubed number z, then z^3=2x+1 and x=(z^3-1)/2. Now consider the cube of an even number. The difference between the squares of integer x and x+2 is 4(x+1), an even number. So we can write 4(x+1)=z^3, and x=(z^3-4)/4. Therefore it is is always possible to find two integers the difference of whose squares is equal to the cube of an integer.