Kenneth Rosen Edition 7 Exercise 1.7 Question 7 (Page No. 91)

Question

Use a direct proof to show that every odd integer is the difference of two squares.

Answer 1

We know that every odd number can be expressed in the form of $2k+1$ where $k$ is a whole number. Adding and subtracting $k^2$ from $2k+1$ gives us $k^2 + 2k + 1 - k^2 = (k+1)^2 - k^2 = 2k+1$. This means that every odd number of the form $2k+1$ can be expressed as the difference of $(k+1)^2$ and $k^2$ for all whole numbers $k$.