# Talk:Coprime

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 Definition:  Integers, or more generally elements of a ring, which have no non-trivial common factor. [d] [e]
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## Main Entry: Coprime or Relatively Prime

These are usually synonymous, so either could be used as the main entry on this subject. However, I wonder if they are synonymous in the following context:

Does one ever say that a set of integers with more than 2 elements are coprime to each other? I do not mean in the pairwise sense, where I believe the terms "pairwise coprime" and "pairwise relatively prime" are totally synonymous.

One would say that 6,10, and 15 are relatively prime to each other, albeit not pairwise. Would anyone ever say that 6, 10, and 15 are coprime to each other? In my mind, the prefix "co-" which is ubiquitous in higher mathematics conjures up the idea of a duality, i.e., dimension versus codimension, domain versus codomain, etc. This also seems to me to apply when one speaks of 5 and 6 as being coprime -- write down the two prime factorizations, and observe that no prime appearing in one can appear in the other.

Am I wrong in having this linguistic view of the prefix "co-"? The problem with the above example is that when one says 6, 10, and 15 are relatively prime to each other, one cannot analyze this statement without examining all three integers simultaneously -- duality does not arise in this consideration. As such, it seems to me that the term "relatively prime" is more general and should be the main entry heading.

Any thoughts? ...said Barry R. Smith (talk) 13:19, 30 October 2008

I don't believe that 'co' implies any sort of duality here. I would say that a set of three integers are coprime if there is no single factor common to all of them: coprime in pairs if any two of them are coprime; eg {6,10,15}. Richard Pinch 07:24, 31 October 2008 (UTC)