Coprime integers

In mathematics, two integers a and b are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1.[1] Consequently, any prime number that divides a does not divide b, and vice versa. This is equivalent to their greatest common divisor (GCD) being 1.[2] One says also a is prime to b or a is coprime with b.

The numbers 14 and 25 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 14 and 21 are not coprime, because they are both divisible by 7. The numerator and denominator of a reduced fraction are coprime, by definition.

A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm and its faster variants such as binary GCD algorithm or Lehmer's GCD algorithm.

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

The numbers 1 and −1 are the only integers coprime with every integer, and they are the only integers that are coprime with 0.

As a consequence of the third point, if a and b are coprime and brbs (mod a), then rs (mod a).[5] That is, 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 (i.e., modulo a it is a product of invertible elements, and therefore invertible);[6] 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 ak and bm.

If a and b are coprime and a divides the product bc, then a divides c.[7] This can be viewed as a generalization of Euclid's lemma.

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

The two integers a and b are coprime if and only if the point with coordinates (a, b) in a Cartesian coordinate system would be "visible" via an unobstructed line of sight from the origin (0,0), in the sense that there is no point with integer coordinates anywhere on the line segment 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, which is about 61% (see § Probability of coprimality, below).

Two natural numbers a and b are coprime if and only if the numbers 2a − 1 and 2b − 1 are coprime.[8] As a generalization of this, following easily from the Euclidean algorithm in base n > 1:

If every pair in a set of integers is coprime, then the set is said to be pairwise coprime (or pairwise relatively prime, mutually coprime or mutually relatively prime). Pairwise coprimality is a stronger condition than setwise coprimality; every pairwise coprime finite set is also setwise coprime, but the reverse is not true. For example, the integers 4, 5, 6 are (setwise) coprime (because the only positive integer dividing all of them is 1), but they are not pairwise coprime (because gcd(4, 6) = 2).

The concept of pairwise coprimality is important as a hypothesis in many results in number theory, such as the Chinese remainder theorem.

It is possible for an infinite set of integers to be pairwise coprime. Notable examples include the set of all prime numbers, the set of elements in Sylvester's sequence, and the set of all Fermat numbers.

Two ideals A and B in a 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 = AB; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem can be generalized to any commutative ring, using coprime ideals.

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).

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.

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. There are coprime pairs near the axes and in some of the gaps but with dots too small to see here.

In machine design, an even, uniform gear wear is achieved by choosing the tooth counts of the two gears meshing together to be relatively prime. When a 1:1 gear ratio is desired, a gear relatively prime to the two equal-size gears may be inserted between them.

In pre-computer cryptography, some Vernam cipher machines combined several loops of key tape of different lengths. Many rotor machines combine rotors of different numbers of teeth. Such combinations work best when the entire set of lengths are pairwise coprime.[12][13][14][15]

This concept can be extended to other algebraic structures than ; for example, polynomials whose greatest common divisor is 1 are called coprime polynomials.