# Why is 1 not a prime number?

**URL:** <https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549>\
**Category:** Factual Questions\
**Created:** [April 14, 2019, 7:57pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549 "2019-04-14T19:57:40Z")\
**Posts on this page:** 5\
**Page:** 4

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**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\
**Post date:** [April 28, 2019, 3:03am UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/64 "2019-04-28T03:03:06Z")

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No, but 1 - 2i is equivalent to 2 + i, in that they both divide each other (multiply by i to go from left to right, or by -i to go from right to left), and similarly, 1 + 2i is equivalent to 2 - i (just the conjugates of before). When people talk about unique prime factorization in these contexts, they always mean up to such equivalence, which is to say, modulo multiplication by units.

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**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\
**Post date:** [April 28, 2019, 12:22pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/65 "2019-04-28T12:22:34Z")

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OK, that’s sufficiently obvious that I really should have noticed it.

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**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\
**Post date:** [April 28, 2019, 2:23pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/66 "2019-04-28T14:23:19Z")

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Actually, the easiest (but not the simplest since I would have to define several terms) way to state unique factorization in a ring R without zero divisors, whose whose subset R\* of non-zero elements thereby forms a monoid, and whose group of units is U, is that R\*/U is free monoid. The generators are the equivalence classes of) primes. You do that and 1 is not a prime. And 0 is ignored. Note that while generally cannot form the quotient of a monoid by a submonoid, you can when the submonoid is a group, which the units always are essentially by definition of unit.

From that point of view the question would never arise.

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**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\
**Post date:** [April 28, 2019, 3:46pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/67 "2019-04-28T15:46:48Z")

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Free _commutative_ monoid, rather.

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**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\
**Post date:** [April 29, 2019, 10:38pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/68 "2019-04-29T22:38:18Z")

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> [@Indistinguishable](#):
>
> Free _commutative_ monoid, rather.

Absolutely right, sorry I misspoke.

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