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I'm writing something vaguely mathematical and I'd like to discuss a particular concept but I could do with a snappy word or phrase for this idea.

All natural numbers have an associated list of factors. (For example, 12's is 1,2,3,4,6 and 12.) I'm looking to talk about a number's factors, but not including itself and 1. Does that have a less wordy name than "factors that aren't itself and one"?

What is the thing that prime numbers definitionally do not have, but composite numbers do have?

Gonçalo
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billpg
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    "proper divisors" is sometimes used. – lulu Mar 11 '25 at 12:22
  • I know the term "real divisors" (in Dutch "echte deler"), but that only loses the number $1$ (as it's a divisor of all numbers). – Dominique Mar 11 '25 at 12:28
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    @lulu But "proper divisor" usually includes $1$ (and all units) whereas "nontrivial" excludes units. $\ \ $ – Bill Dubuque Mar 11 '25 at 12:59
  • My two cents - I think the terminology that is most likely to speak for itself and not require further clarification is "proper non-trivial divisor". I think anyone would interpret "proper" to mean at least "not equal to $n$" and "non-trivial" to mean at least "not equal to $1$". This terminology is analogous to "proper non-trivial subset/subgroup/subspace" etc which are fairly commonplace. – Izaak van Dongen Mar 11 '25 at 13:00
  • @BillDubuque I think I was never clear on that usage. Looking here say, confirms your usage, so I agree. Thanks. – lulu Mar 11 '25 at 13:00
  • See the first paragraph in the linked dupe. The units (invertibles) of $,\Bbb N,$ and $,\Bbb Z,$ are $1$ and $\pm 1,,$ resp. $\ \ $ – Bill Dubuque Mar 11 '25 at 13:16
  • This is not a duplicate of the question linked. I barely understand that question, but I understand enough to see it isn't a question about terminology. – billpg Mar 11 '25 at 13:49
  • Said first paragraph defines a trivial factor of $,n,$ as either a unit $(\pm1)$ or a unit multiple $(\pm n),\ $ [omit negatives for $\Bbb N]., $ Any factor that is not trivial is called a nontrivial factor. Another example: for Gaussian integers $,\alpha = a+bi,\ a,b\in\Bbb Z,,$ the units $,u,$ are $,i^n = \pm 1,\pm i ,$ so the trivial factors are those of the form $,u,$ and $,u:!\alpha,,$ i.e. units, and unit multiples of $,\alpha.\ \ $ – Bill Dubuque Mar 11 '25 at 14:28
  • In $,\Bbb Q[x] = $ ring of all polynomials with rational coef's, the units (invertibles) are all nonzero rationals, so the trivial factors of $,f(x),$ are $,q,$ and $,q:! f(x),,$ for $,q,$ any rational. $\ \ $ – Bill Dubuque Mar 11 '25 at 14:29

2 Answers2

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You could use "non-trivial divisors". Note that for all $n$, $1|n$ and $n|n$ so you can refer to $1$ and $n$ as the "trivial divisors" of $n$. Hence the rest are the non-trivial ones

Najdorf
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As others suggest, proper or non-trivial are good terms but neither will be certainly interpreted correctly. There is no international standard body setting mathematical terminology. So, pick one and say what it means.

badjohn
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  • Please strive not to post more (dupe) answers to dupes of FAQs. This is enforced site policy, see here. – Bill Dubuque Mar 11 '25 at 13:17
  • It's hard for me to see how the meaning of "nontrivial divisor of a natural number" can be interpreted incorrectly. – MPW Mar 11 '25 at 13:25