Pairwise coprime

In mathematics, especially number theory, a set of integers is said to be pairwise coprime (or pairwise relatively prime, also known as mutually coprime) if every pair of distinct integers a and b in the set are coprime (that is, have no common divisors other than 1). The concept of pairwise coprimality is important in applications of the Chinese remainder theorem and the proof that x³ + y³ + z³ = 0 has no nonzero integer solutions.

Definition

A set P of integers is pairwise coprime iff, for every p and q in P with p ≠ q, we have gcd(p, q) = 1. Here gcd denotes the greatest common divisor.

Examples

The set {10, 7, 33, 13} is pairwise coprime, because any pair of the numbers have a Greatest common divisor equal to 1:

gcd(10, 7) = gcd(10, 33) = gcd(10, 13) = gcd(7, 33) = gcd(7, 13) = gcd(33, 13) = 1.

Where gcd(a, b) refers to the greatest common divisor of a and b.

On the other hand, the integers 10, 7, 33, 14 are not pairwise coprime, because gcd(10, 14) = 2 ≠ 1 and gcd(7, 14) = 7 ≠ 1.

Infinite examples

The set of all primes, {2, 3, 5, 7, 11, …} is of course pairwise coprime, as is the set of elements in Sylvester's sequence, and the set of all Fermat numbers.

Usage

It is permissible to say "the integers 10, 7, 33, 13 are pairwise coprime", rather than the more exacting "the set of integers {10, 7, 33, 13} is pairwise coprime".

"Pairwise coprime" vs "coprime"

The concept of pairwise coprimality is stronger than that of coprimality. The latter indicates that the greatest common divisor of all integers in the set is 1. For example, the integers 6, 10, 15 are coprime (because the only positive integer dividing all of them is 1), but they are not pairwise coprime because the greatest common divisor or gcd of (6, 10) = 2, gcd of (10, 15) = 5 and gcd of (6, 15) = 3. On the other hand if some integers are pairwise coprime then they are certainly coprime, i.e. pairwise coprimality implies coprimality but not vice versa. To prove the implication it is sufficient to note that any common divisor of all the integers can only be 1 (otherwise pairwise coprimality will be violated).