Pairwise coprime

In mathematics, especially number theory, a set of integers is said to be pairwise coprime (or pairwise relatively prime, mutually coprime or mutually relatively prime) if every two distinct integers a and b in the set are coprime (that is, have no common positive divisors other than 1).

Pairwise coprimality should be contrasted with coprimality (unmodified by the word "pairwise"; also known as setwise coprimality), in which it is requested only that no positive integer other than 1 divides all the members of the set.

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

Examples

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

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

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 is pairwise coprime, as is the set of elements in Sylvester's sequence, and the set of all Fermat numbers.

"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 gcd(6, 10) = 2, gcd(10, 15) = 5 and gcd(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).

Alternative definition

A set P of integers is pairwise coprime if and only if there is no prime number that divides two or more elements of P.