Pairwise coprime: Difference between revisions
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#REDIRECT [[Coprime integers#Coprimality in sets]] |
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In [[mathematics]], especially [[number theory]], a set of [[integer]]s 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 [[divisor]]s other than 1). The concept of pairwise coprimality is important in applications of the [[Chinese remainder theorem]] and the proof that x<sup>3</sup> + y<sup>3</sup> + z<sup>3</sup> = 0 has no nonzero integer solutions. |
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== Definition == |
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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]]. |
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== Examples == |
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The set {10, 7, 33, 13} is pairwise coprime, because any pair of the numbers have a Greatest common divisor equal to 1: |
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: gcd(10, 7) = gcd(10, 33) = gcd(10, 13) = gcd(7, 33) = gcd(7, 13) = gcd(33, 13) = 1. |
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Where gcd(a, b) refers to the greatest common divisor of a and b. |
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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. |
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=== Infinite examples === |
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The set of all primes is of course pairwise coprime, as is the set of elements in [[Sylvester's sequence]], and the set of all [[Fermat numbers]]. |
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== "Pairwise coprime" vs "coprime" == |
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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). |
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== Alternative definition == |
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A set ''P'' of integers is '''pairwise coprime''' [[iff]], considering their prime factorization, there is no factor common to two or more of them. |
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{{numtheory-stub}} |
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[[Category:Number theory]] |
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[[nl:Paarsgewijs relatief priem]] |
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Latest revision as of 00:36, 15 June 2017
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