Talk:Integers Coprime to Zero

The fact is the situation where both $m$ and $n$ are zero is covered, at the bottom of the page.

As for the other point, I don't know what the problem is. --prime mover (talk) 09:49, 3 September 2022 (UTC)

Sorry, where is the situation defined? Especially, where is $0 \perp 0$ defined? We need a correct citation.

The other point is similar. What should be $\gcd \set{n,0}$ is $n=0$? Where is this defined?--Usagiop (talk) 10:23, 3 September 2022 (UTC)

We need to assume $n \in \Z_{\ne 0}$.--Usagiop (talk) 10:25, 3 September 2022 (UTC)

Definition:Coprime Integers defines when $m,n$ are coprime for $m \ne 0$ or $n \ne 0$. I would argue that since it is not defined what $0$ is comprime to $0$ should mean, it follows that $0$ is not coprime to $0$. I understand that the suggestion is we should not write $n \perp 0$ while letting $n \in \Z$, but I do not think it is wrong.

Second, the proof directly mentions that $\gcd \set {n, 0}$ is not defined. --Anghel (talk) 11:21, 3 September 2022 (UTC)

It's not defined only when $n = 0$. As $0 \notin \set {1, -1}$ that does not invalidate the proof.

Sorry, I meant to write $\gcd \set {n, 0}$ is not defined when $n=0$. Of course you are right here. I have no more comments to the statement of this theorem. --Anghel (talk) 23:15, 3 September 2022 (UTC)

$\Leftarrow$ is OK. But when you consider $\Rightarrow$, you need to handle the case $n=0$.--Usagiop (talk) 23:36, 3 September 2022 (UTC)

I got your point, but the exposition should be improved. This is very hard to understand. In your opinion, the following claim is also true:

$n \in \C: n \perp 0 \iff n \in \set {1, -1}$

I would say this is wrong.--Usagiop (talk) 17:24, 3 September 2022 (UTC)

What specifically is "wrong" about it? $0 \perp 0$ is undefined so by definition $0 \perp 0 \notin \set {1, -1}$. Unless the thinking behind that statement is that the na\"ive interpretation of "undefined" does not exclude it from being in $\set {1, -1}$ I'm afraid you've lost me. --prime mover (talk) 21:35, 3 September 2022 (UTC)

Maybe you could be right, but such an exposition is unfriendly. You should define it for all cases. Undefined is undefined, not equal to the logical "False". In addition, I can never be sure if it is really undefined or it is just my fault that I cannot find the definition anywhere.--Usagiop (talk) 22:29, 3 September 2022 (UTC)

"You should define it for all cases." So if the problem is on the page defining coprimality, or even GCD, then maybe that's the page where you should be raising your concerns. --prime mover (talk) 07:30, 4 September 2022 (UTC)

Thank you for the change in Definition:Coprime Integers. Now this page looks fine.--Usagiop (talk) 10:06, 4 September 2022 (UTC)

I have raised the page Coprime Integers cannot Both be Zero which makes this explicit. If this is insufficient to satisfy your worries, then I'm afraid you may need help which I cannot provide. --prime mover (talk) 08:06, 4 September 2022 (UTC)