Even square implies even number

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

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

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

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

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Any odd number squared equals 2(2n² + 2n) + 1, and let k = 2n² + 2n.

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Any odd number squared equals 4n² + 4n + 1.

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

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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.

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