What does $(m, n) = 1$ mean?

Question

I have to solve a problem which states that $(m,n) = 1$, but I have no idea what this means. Maybe, the problem itself will help:

Suppose $m, n \in \mathbb{Z}$, with $n > 0$ and $(m,n) = 1$. Show that:

$$\left \{ e^{2 \pi i m k/n}: 0 \leq k < n \right \} = \left \{ e^{2 \pi i j/n}: 0 \leq j < n\right \}$$

Thanks a lot for any help.

I actually think the parenthesis notation is overloaded too much. $(a,b)$ may be a pair, an interval, a gcd … anything else? — celtschk, Apr 30 '15 at 21:31
Well, $(n,m)=1$ clearly overrides the interval and pair cases. — abnry, Apr 30 '15 at 21:35

score 9 · Accepted Answer · edited Apr 13 '17 at 12:21

The notation $\rm\ (a,b)\ $ can denote either $\rm\ gcd(a,b)\ $ or the ideal $\rm\ a\ \mathbb Z + b\ \mathbb Z\:.\ $ In $\rm\:\mathbb Z\:$ (or in any Bezout domain $\rm\:Z\:$) these denote essentially the same object since $\rm\ a\ Z + b\ Z\ =\ c\ Z\ \ \iff\ \ c = gcd(a,b)\:.$ The advantage of the notation is that it allows one to simultaneously prove results for both gcds and ideals by using only those laws that hold true for both. For example, see my post on the Freshman's dream $\rm\ (a,b)^n = (a^n,b^n)\ $ for gcds and invertible ideals. Such analogies are exploited to the hilt when one studies divisor theory.

score 6 · Answer 2 · answered Mar 14 '11 at 15:41

6

$(m,n)=1$ simply means that the G.C.D of $m,n$ is $1$. In other words they are relatively prime.

answered Mar 14 '11 at 15:41

Steven Alexis Gregory · Answer 3 · 2015-04-30T21:34:00.400

The Pidgeon hole principle says that if you have N pidgeons, M pidgeon holes to put them into, and N > M, then at least one pidgeon hole will contain more than one pidgeon.

This effect happens in mathematics a lot. There are more mathematical concepts than there are mathematical symbols. So we have to use the same symbols to denote different concepts.

In your case, (m, n) means the GCD of m and n. It could also mean

The point with coordinates (m, n).
The interval (m, n).
The smallest ideal containing m and n.

and I'm sure there are other meanings. Usually mathematicians justify why they don't explain which concept they mean by saying that "the context should make it obvious".

When I see a notation that I don't understand, I assume one of the following

They explained it and I just didn't see it.
They are pompous ****s who don't want to waste their time on people who aren't as smart as they are.
They just can't understand how anyone could be confused with such a well-written article.

What does $(m, n) = 1$ mean?

3 Answers3