# Number Theory | No Common Factor Notation?

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

• 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. – Robert Soupe Sep 3 '16 at 2:04
• I've no problem with meaning. I just want to make writing shorter with notations. – Gal Grünfeld Sep 4 '16 at 1:29
• 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. – Robert Soupe Sep 4 '16 at 4:56
• 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. – Gal Grünfeld Sep 5 '16 at 23:22
• 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. – Robert Soupe Sep 6 '16 at 0:40

let $$a,b$$ are integer numbers, then we can write GCD(a,b)=1