# I can't find what't wrong with this proof, it's from a discrete mathematics slide

This is a proof in Discrete Mathematics.

• Please include the proof in the text to make it accessible to more people. Also, what are your thoughts? – Carsten S Jul 27 '17 at 12:52
• @CarstenS He already included his thoughts: "I can't find what it is wrong". – Peyton Jul 27 '17 at 12:53
• The problem is with the $\leftarrow$. That you would need $a^2$ to be the double of an even number plus $1$. Therefore that implication is still missing proving that if $a^2$ is odd, it should have that form. – Peyton Jul 27 '17 at 12:54
• After the third $\leftrightarrow$, it says $4k^4 + 4\color{red}{K} + 1$. Should be $4k^2 + 4\color{red}{k} + 1$. Nailed it! – user307169 Jul 27 '17 at 12:55
• @tilper LOL!!!! – Peyton Jul 27 '17 at 12:56

The problem lies in the spurious use of biimplications. If we read the left-to-right implications, we get a correct proof of the statement

If $a$ is odd, then $a^2$ is odd

However, the reverse implication is not correct. We cannot conclude that if $a^2$ is odd, then $a^2 = 2(2k^2+2k)+1$ for some $k \in \mathbb{N}$. All we know is that then $a^2 = 2K+1$ for some $K \in \mathbb{N}$.

• Well, yes, you can conclude that, although one could argue that there are steps missing in the proof. – Dirk Jul 27 '17 at 13:09

The problem is when you go back from

$a^2$ is odd

To

$a^2 = 2(2k^2+2k)+1$

That does not (immediately) follow!

To be correct, you have to write

Let $n\in \Bbb N$.

$n$ is odd $\implies \exists k\in \Bbb N \;\; n=2k+1$

$$\implies \exists k\in \Bbb N \;: n^2=2 (2k^2+2k)+1$$

$$\implies \exists K=2 (2k^2+2k)\in \Bbb N \; :$$ $$n^2=2K+1$$ $\implies n^2$ is odd.

Conversly,

$n$ is even $\implies \exists k\in \Bbb N \; : n=2k$

$$\implies \exists k\in\Bbb N \; : n^2=4k^2$$

$$\implies \exists K=2k^2\in \Bbb N \;: n^2=2K$$ $\implies n^2$ is even.

we conclude that $n$ is odd $\iff n^2$ is odd.

It should be: $a$ is odd $\iff a=2k-1, k\in N.$

• This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review – Alberto Debernardi Jul 27 '17 at 14:19
• @AlbertoDebernardi, thank you for advice. I think this is the seriuos flaw in the proof, since it leaves out the odd number $1$. – farruhota Jul 27 '17 at 15:27