Definition:Coprime

Let $$a$$ and $$b$$ be integers such that $$b \ne 0$$ and $$a \ne 0$$ (i.e. they are not both zero).

Let $$\gcd \left\{{a, b}\right\}$$ be the greatest common divisor of $$a$$ and $$b$$.

If $$\gcd \left\{{a, b}\right\} = 1$$, then $$a$$ and $$b$$ are coprime, or relatively prime.

Alternatively we can say $$a$$ is prime to $$b$$.

If $$\gcd \left\{{a, b}\right\} = 1$$, then the notation $$a \perp b$$ is encouraged.

If $$\gcd \left\{{a, b}\right\} \ne 1$$, the notation $$a \not \perp b$$ can be used.

It can be seen that considered as a relation, $$\perp$$ is:


 * 1) Non-reflexive: $$a \not \perp a$$ except when $$a = \pm 1$$;
 * 2) Symmetric: $$a \perp b \iff b \perp a$$;
 * 3) Not antisymmetric: $$a \perp b \land b \perp a \not \implies a = b$$;
 * 4) Non-transitive: Consider $$2 \perp 3, 3 \perp 4, 2 \not \perp 4$$ and $$2 \perp 3, 3 \perp 5, 2 \perp 5$$.

Comment
"This concept is so important in practice, we ought to have a special notation for it; but, alas, number theorists haven't agreed on a very good one yet. Therefore we cry: "HEAR US, O MATHEMATICIANS OF THE WORLD! LET US NOT WAIT ANY LONGER! WE CAN MAKE MANY POPULAR FORMULAS CLEARER BY ADOPTING A NEW NOTATION NOW! LET US AGREE TO WRITE '$$m \perp n$$' AND TO SAY "$$m$$ is prime to $$n$$," IF $$m$$ AND $$n$$ ARE RELATIVELY PRIME."

Can't say it any clearer than that.