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I have a question about notation in number theory:

Is there a notation for a set of integers to not have a common factor?

Maybe something like: $\neg\exists\, gcd(\{z \in \mathbb{Z}\,|\,whatever\})$?

Edit: where $gcd(my set) \not=1$

I'm looking for notation for writing a proof that $\sqrt{3} \notin \mathbb{Q} \,$ (i.e, it's irrational).

*Note: I didn't learn number theory. This proof I want to do is an exercise from a preparation lesson to calculus (about axioms of $\mathbb{R}$eal numbers and a few other things).

Thanks a lot to the helpers in advance! :D

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  • $\begingroup$ What about just saying "these numbers are all pairwise coprime"? People get carried away with notation and then forget what they meant in the first place. $\endgroup$ – Robert Soupe Sep 3 '16 at 2:04
  • $\begingroup$ I've no problem with meaning. I just want to make writing shorter with notations. $\endgroup$ – Gal Grünfeld Sep 4 '16 at 1:29
  • $\begingroup$ What about the people who then have to decode those short but dense lines? e.g., here's a well-known number-theoretic function: $\mu(n) = \delta_{\omega(n)}^{\Omega(n)} (-1)^{\omega(n)}$, can you tell what it is? Hint, it's more commonly defined with a brace for three cases. $\endgroup$ – Robert Soupe Sep 4 '16 at 4:56
  • $\begingroup$ So does this generalize a definition of the function $u(n$), that, not in this form, is written in 3 cases? Do you think that I know what $u(n)$ is? I barely know anything in number theory, FYI. $\endgroup$ – Gal Grünfeld Sep 5 '16 at 23:22
  • $\begingroup$ It's the Möbius mu function. If you don't know it already, you will soon if you keep studying number theory. The point is that conciseness sometimes goes against clarity. $\endgroup$ – Robert Soupe Sep 6 '16 at 0:40
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let $$a,b$$ are integer numbers, then we can write GCD(a,b)=1

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  • $\begingroup$ Awesome, thank you very much. :) $\endgroup$ – Gal Grünfeld Sep 2 '16 at 11:08
  • $\begingroup$ Wait... what!!??? $\endgroup$ – Robert Soupe Sep 3 '16 at 2:04
  • $\begingroup$ Well... I don't think the OP typed so much just to read this. Surprising is that he is convinced too. :3 $\endgroup$ – I am Back Apr 11 '17 at 20:21

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