Relatively Prime

Relatively prime integers share no common positive divisor except 1. Using the notation to denote the greatest common divisor, two integers and are relatively prime iff . Relatively prime integers are sometimes also called strangers or coprime and are denoted . The plot above plots and along the two axes and colors a square black if and white otherwise (left figure) and simply colored according to (right figure).

Two integers can be tested to see if they are relatively prime in the Wolfram Language using CoprimeQ[m, n].

Two distinct primes and are always relatively prime, , as are any positive integer powers of distinct primes and , .

Relative primality is not transitive. For example, and , but .

For two independently and uniformly chosen positive integers and from , the limiting probability as that they are relatively prime is

(1)

(OEIS A059956; Cesàro and Sylvester 1883, Lehmer 1900, Sylvester 1909, Nymann 1972; Wells 1986, p. 28; Borwein and Bailey 2003, p. 139; Havil 2003, pp. 40 and 65; Moree 2005), where is the Riemann zeta function. This result is related to the fact that the greatest common divisor of and , , can be interpreted as the number of lattice points in the plane which lie on the line segment connecting the vectors and (excluding itself). In fact, is the natural density of lattice points with positive integer coordinates visible from the origin (Castellanos 1988, pp. 155-156).

Similarly, for three independent positive integers , , and sampled uniformly from , the limiting probability as that their common greatest common divisor is 1 is

(2)

(OEIS A088453; Wells 1986, p. 29), where is Apéry's constant (Wells 1986, p. 29). In general, for independent positive integers sampled in this way and a positive integer with , the limiting probability that no common divisor is a th power with integer is (Cohen 1959, Salamin 1972, Nymann 1975, Schoenfeld 1976, Porubský 1981, Chidambaraswamy and Sitaramachandra Rao 1987, Hafner et al. 1993).

For Gaussian integers, two elements are relatively prime when all their common divisors are units. For two independently and uniformly chosen Gaussian integers and of complex modulus at most , the limiting probability as that they are relatively prime is

(3)

(OEIS A088454), where is Catalan's constant (Collins and Johnson 1989, Pegg 2003; Finch 2003, p. 601).

Similarly, for two independently and uniformly chosen Eisenstein integers of complex modulus at most , the limiting probability as that their only common divisors are units is

(4)

(OEIS A088467), where

(5)

(Finch 2003, p. 601), which can be written analytically as

			(6)
			(7)

(OEIS A086724), where is the trigamma function.

The limiting probabilities for pairs of integers and Gaussian integers being relatively prime also equal the natural densities of squarefree elements of the respective rings.