Coprime integers

In number theory, a branch of mathematics, two integers a and b are said to be coprime (also spelled co-prime), relatively prime or mutually prime if the only positive integer that evenly divides both of them is 1. This is equivalent to their greatest common divisor being 1. In addition to $\gcd(a, b) = 1\;$ and $(a, b) = 1,\;$ the notation $a\perp b$ is sometimes used to indicate that a and b are relatively prime.

For example, 14 and 15 are coprime, being commonly divisible by only 1, but 14 and 21 are not, because they are both divisible by 7. The numbers 1 and −1 are coprime to every integer, and they are the only integers to be coprime with 0.

A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm.

The number of integers coprime to a positive integer n, between 1 and n, is given by Euler's totient function (or Euler's phi function) φ(n).

Properties

Figure 1. The numbers 4 and 9 are coprime. Therefore, the diagonal of a 4 x 9 lattice does not intersect any other lattice points

There are a number of conditions which are equivalent to a and b being coprime:

• No prime number divides both a and b.
• There exist integers x and y such that ax + by = 1 (see Bézout's identity).
• The integer b has a multiplicative inverse modulo a: there exists an integer y such that by ≡ 1 (mod a). In other words, b is a unit in the ring Z/aZ of integers modulo a.
• Every pair of congruence relations for an unknown integer x, of the form x ≡ k (mod a) and x ≡ l (mod b), has a solution, as stated by the Chinese remainder theorem; in fact the solutions are described by a single congruence relation modulo ab.

As a consequence of the third point, if a and b are coprime and br ≡ bs (mod a), then r ≡ s (mod a) (because we may "divide by b" when working modulo a). Furthermore, if b₁ and b₂ are both coprime with a, then so is their product b₁b₂ (modulo a it is a product of invertible elements, and therefore invertible); this also follows from the first point by Euclid's lemma, which states that if a prime number p divides a product bc, then p divides at least one of the factors b, c.

As a consequence of the first point, if a and b are coprime, then so are any powers a^k and b^l.

If a and b are coprime and a divides the product bc, then a divides c. This can be viewed as a generalization of Euclid's lemma.

The two integers a and b are coprime if and only if the point with coordinates (a, b) in a Cartesian coordinate system is "visible" from the origin (0,0), in the sense that there is no point with integer coordinates between the origin and (a, b). (See figure 1.)

In a sense that can be made precise, the probability that two randomly chosen integers are coprime is 6/π² (see pi), which is about 61%. See below.

Two natural numbers a and b are coprime if and only if the numbers 2^a − 1 and 2^b − 1 are coprime. As a generalization of this, following easily from Euclidean algorithm in base n > 1:

$\gcd (n^a-1,n^b-1)=n^{\gcd(a,b)}-1.$

Cross notation, group

See also: multiplicative group of integers modulo n

If n≥1 and is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation; it is written as (Z/nZ)^× or Z_n^*.

Generalizations

The concept of being relatively prime can also be extended to any finite set of integers S = {a₁, a₂, .... a_n} to mean that the greatest common divisor of the elements of the set is 1. If every pair in a (finite or infinite) set of integers is relatively prime, then the set is called pairwise relatively prime. Every pairwise relatively prime finite set is relatively prime; however, the converse is not true: {6, 10, 15} is relatively prime (the only positive integer that divides all of 6, 10 and 15 is 1), but not pairwise relative prime (each pair of integers in the set has a non-trivial common factor).

Two ideals A and B in the commutative ring R are called coprime (or comaximal) if A + B = R. This generalizes Bézout's identity: with this definition, two principal ideals (a) and (b) in the ring of integers Z are coprime if and only if a and b are coprime.

If the ideals A and B of R are coprime, then AB = A∩B; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem is an important statement about coprime ideals.

Probabilities

Given two randomly chosen integers a and b, it is reasonable to ask how likely it is that a and b are coprime. In this determination, it is convenient to use the characterization that a and b are coprime if and only if no prime number divides both of them (see Fundamental theorem of arithmetic).

Informally, the probability that any number is divisible by a prime (or in fact any integer) $p$ is $1/p$; for example, every 7th integer is divisible by 7. Hence the probability that two numbers are both divisible by this prime is $1/p^2$, and the probability that at least one of them is not is $1-1/p^2$. For distinct primes, these divisibility events are mutually independent. For example, in the case of two events, a number is divisible by p and q if and only if it is divisible by pq; the latter event has probability 1/pq. (Independence does not hold for numbers in general, but holds for prime numbers.) Thus the probability that two numbers are coprime is given by a product over all primes,

$\prod_p^{\infty} \left(1-\frac{1}{p^2}\right) = \left( \prod_p^{\infty} \frac{1}{1-p^{-2}} \right)^{-1} = \frac{1}{\zeta(2)} = \frac{6}{\pi^2} \approx 0.607927102 \approx 61\%.$

Here ζ refers to the Riemann zeta function, the identity relating the product over primes to ζ(2) is an example of an Euler product, and the evaluation of ζ(2) as π²/6 is the Basel problem, solved by Leonhard Euler in 1735. In general, the probability of k randomly chosen integers being coprime is 1/ζ(k).

The notion of a "randomly chosen integer" in the preceding paragraphs is not rigorous. One rigorous formalization is the notion of natural density: choose the integers a and b randomly between 1 and an integer N. Then, for each upper bound N, there is a probability P_N that two randomly chosen numbers are coprime. This will never be exactly $6/\pi^2$, but in the limit as $N \to \infty$, the probability $P_N$ approaches $6/\pi^2$.

Generating all coprime pairs

The order of generation of coprime pairs by this algorithm. First node (2,1) is marked red, its three children are shown in orange, third generation is yellow, and so on in the rainbow order.

All pairs of coprime numbers $m, n$ can be arranged in a pair of disjoint complete ternary trees, starting from $(2,1)$ (for even-odd or odd-even pairs) or from $(3,1)$ (for odd-odd pairs). The children of each vertex $(m,n)$ are generated as follows:

Branch 1: $(2m-n,m)$

Branch 2: $(2m+n,m)$

Branch 3: $(m+2n,n)$

This scheme is exhaustive and non-redundant with no invalid members.

Further reading

• Lord, Nick (March 2008), "A uniform construction of some infinite coprime sequences", Mathematical Gazette 92: 66–70 .