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.