What's the meaning of this $(m,n) = 1$

Question

I'm reading this pdf http://rutherglen.science.mq.edu.au/wchen/lndpnfolder/dpn01.pdf I understand some of the expression used in this but I don't understand the part $(m,n) = 1$

Is this a cartesian coordinate or some sort of operation?

Answer 1

The notation $\,(a,b) := \gcd(a,b)\,$ is widely used in number theory. Similarly, but less frequently, authors use $\,\ \ [a,b]\, := {\rm lcm}(a,b).\,$ Hence $\,(a,b) = 1\,$ means $\,a,b\,$ are coprime: $\,c\mid a,b\,\Rightarrow\,c\mid 1.$

Here, as often, one can uniquely infer the meaning from its use. The first use of the notation is in Theorem $1$ where it is claimed that $\,(a,b)=1\iff $ every positive divisor of $\,ab\,$ can be written uniquely in the form $\,cd\,$ where $\,c\mid a,\ d\mid b,\,\ c,d\in\Bbb N.\,$ This implies that $\,a,b\,$ are coprime since, if not, they have a common divisor $\,n> 1$ which has more than one such rep: $\,c,d = n,1\,$ and $\,1,n.$

One of the reasons that the gcd tuple notation is so widely used is that it helps to emphasize analogies between gcds and ideals, which use the same tuple notation. They share many of the same arithmetical laws, e.g. associative, distributive, and $\,(a,b) = (a)\,$ if $\,a\mid b,\,$ so by using a common notation one can write proofs that work for both gcds and ideals. In PIDs like $\,\Bbb Z\,$ they are essentially arithmetically equivalent, since for ideals $\ (a,b) = (c) \iff c = \gcd(a,b),\,$ so one can read $\,(a,b)\,$ either as a gcd or an ideal. The analogies between ideals and gcds are clarified when one studies divisor theory (the modern version of Kronecker's alternative to Dedekind's ideal-theoretic approach to factorization in rings of algebraic integers). An introduction to divisor theory can be found in Borevich and I. R. Shafarevich, Number theory. See also

Friedemann Lucius. Rings with a theory of greatest common divisors.
manuscripta math. 95, 117-36 (1998).

Olaf Neumann. Was sollen und was sind Divisoren?
(What are divisors and what are they good for?) Math. Semesterber, 48, 2, 139-192 (2001).

Because there is little written in English on divisor theory, it would be very valuable to translate Neumann's very nice German exposition into English. If anyone is interested in doing so please contact me.

Answer 2

It means that the greatest common divisor of $m$ and $n$, which is the largest integer dividing both of $m$ and $n$ is equals to $1$.

In otherwords, they are coprime, since $1$ is the largest integer dividing both of them.

Answer 3

$(m, n)$ is used by some to denote $\gcd(m, n),$ the greatest common divisor function of two integers, $m, n$.

So $(m, n) = 1$ means that the greatest common divisor of $m, n$ is $1$; i.e., $m, n$ are relatively prime.

Whether $(m, n)$ is used to mean $\gcd(m, n)$, or an ordered pair, depends on the context in which that notation is used.

Answer 4

$(m,n)$ is the gcd of m and n.