If the square of a number is even, then the number is even.

*If you believe…*

If a number is not even, then the square of that number is not even.

*If you believe…*

If a number is odd, then the square of that number is odd.

*If you believe…*

Any odd number squared equals 2*k* + 1 for some integer *k*.

*If you believe…*

Any odd number squared equals 2(2*n*^{2} + 2*n*) + 1, and let *k* = 2*n*^{2} + 2*n*.

*If you believe…*

Any odd number squared equals 4*n*^{2} + 4*n* + 1.

*If you believe…*

Any odd number equals 2*n* + 1 for some integer *n* (by definition).

## Advisory

*Please note that I am not a mathematician and so the presentation of proofs that I make may be deeply flawed. I'm using this writing process to figure out what I'm reading. Please consult more authoritative sources as well.*

Feel free to contact me by leaving a comment or sending me a private message.