Let p and q be coprime integers, greater than or equal to two.

Then can be written as an irreducible fraction such that and are coprime integers.

More precisely, let and be coprime integers and consider as the unit sphere in .

Otherwise, the "numerator" and "denominator" are now odd positive coprime integers, so we can flip the symbol using rule 6, then return to step 1.

Consider a Fibonacci Box containing two, odd, coprime integers x and y in the right-hand column.

The ring of integers of the field has the following integral basis, writing for two square-free coprime integers h and k:

Let denote the manifold obtained from M by filling in the i-th boundary torus with a solid torus using the slope where each pair and are coprime integers.

These are formed by wrapping a circle around a torus p times in one direction and q times in the other, where p and q are coprime integers.

The naive or classical height of a vector of rational numbers is the maximum absolute value of the vector of coprime integers obtained by multiplying through by a lowest common denominator.

In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n 0.

If are small integers a factorization of the first kind is easy to detect: if with p and q coprime integers, then q divides evenly a, and p divides evenly a. Thus, computing for every possible values of p and q allows to find the rational roots, if any.