# Why isn't 1 a Prime Number? **URL:** \ **Category:** Miscellaneous and Personal Stuff I Must Share\ **Tags:** science-math\ **Created:** [March 30, 2025, 3:55pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346 "2025-03-30T15:55:11Z")\ **Posts on this page:** 11\ **Page:** 3
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [March 31, 2025, 1:19pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/41 "2025-03-31T13:19:08Z")
> [@markn\_1](#): > > Like “threeven” for divisible by 3, “fourven” for divisible by 4, etc. I wonder if any language has something like that. 2: di-visible 3: tri-visible 4: quad-visible 5: pen-visible etc… ---
**Author:** ![solost](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/solost/32/11776_2.png) [@solost](https://boards.straightdope.com/u/solost)\ **Post date:** [March 31, 2025, 1:23pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/42 "2025-03-31T13:23:23Z")
It may not be a _prime_ number, but it’s still #1 in my book. Sorry. I’ll leave the thread now. ---
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [March 31, 2025, 1:25pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/43 "2025-03-31T13:25:35Z")
> [@Jasmine](#): > > I was taught that prime numbers have only two factors, one and itself. Couldn’t some rules-lawyering third-grader give you grief with that definition? Maybe it should be “… prime numbers have **exactly two distinct** factors, one and itself.” ---
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [March 31, 2025, 1:33pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/44 "2025-03-31T13:33:38Z")
> [@Chronos](#): > > Given any positive integer, there is one and only one to express it as a product of primes. Okay, now I have confused myself. How do you express 7 as a product of primes. If you say “7” then I would argue that is not a product. A product requires at least two numbers, at least in my tiny brain. ---
**Author:** ![Jasmine](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jasmine/32/2964_2.png) [@Jasmine](https://boards.straightdope.com/u/Jasmine)\ **Post date:** [March 31, 2025, 2:10pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/45 "2025-03-31T14:10:16Z")
I’m fine with that terminology, but I submit that, in this instance, “only” and “exactly” are synonymous. If someone asked me how many brothers I have, I would say that I had “only one brother”, which I consider a normal conversational response that would clearly convey the correct amount. If said that I have “exactly one distinct” brother, it would convey exactly the same meaning, the only difference being that it would sound kind of strange. ---
**Author:** ![markn\_1](https://avatars.discourse-cdn.com/v4/letter/m/f9ae1b/32.png) [@markn\_1](https://boards.straightdope.com/u/markn_1)\ **Post date:** [March 31, 2025, 3:13pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/46 "2025-03-31T15:13:21Z")
It’s understandable why you would prefer a definition of product that requires at least two factors. But again, it’s a question of usefulness. The standard mathematical definition which allows any number of factors, including one or zero, allows us to say “every integer is a product of primes”. Using your definition, we’d have to say “every integer is either a unit, a prime, or a product of primes”. This is longer and more cumbersome and much more difficult to use in proofs. It makes sense to choose the definition which is more useful, even if it is slightly less intuitive. ---
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [March 31, 2025, 3:42pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/47 "2025-03-31T15:42:33Z")
> [@markn\_1](#): > > … standard mathematical definition which allows any number of factors, including one or zero … To be honest, I knew this was going to be a possible answer before I even finished typing. But I kept typing anyway because I knew this would lead to yet another example of how precise definitions are necessary when discussing math. ---
**Author:** ![leahcim](https://avatars.discourse-cdn.com/v4/letter/l/b4bc9f/32.png) [@leahcim](https://boards.straightdope.com/u/leahcim)\ **Post date:** [March 31, 2025, 5:08pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/48 "2025-03-31T17:08:27Z")
> [@DPRK](#): > > 2 and −2 are prime numbers Often prime numbers are only treated over the natural numbers, since negative primes break unique factorization (e.g. 15 = (5)(3) = (-5)(-3)). AIUI, in more general contexts, ideals are the things that are prime/not prime and negative primes are just different generators for the same ideal. (e.g. 2 and -2 both generate the prime ideal 2Z). ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [April 1, 2025, 10:35am UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/49 "2025-04-01T10:35:19Z")
> [@LSLGuy](#): > > In societies that used 5-based tally marks to record counts of [whatever], I could readily imagine them having a word that meant to them “several complete 5-tallies with no excess items”, but which from our math-centric POV means “divisible by 5”. What about the decimal system everyone uses today? Any word for “divisible by 10”? Actually, I would not be at all surprised if the desired words exist (for at least some base and some language), but we need to find an example. > [@leahcim](#): > > Often prime numbers are only treated over the natural numbers, since negative primes break unique factorization (e.g. 15 = (5)(3) = (-5)(-3)). > > AIUI, in more general contexts, ideals are the things that are prime/not prime and negative primes are just different generators for the same ideal. (e.g. 2 and -2 both generate the prime ideal 2Z). Passing from elements to ideals, multiplying by a unit like −1 will not affect the ideal at all. Now, in fact, historically, working with “ideals” in so-called Dedekind domains solved the problem of unique factorization, because going to ideals does not change divisibility relations but now the ideals in some sense stand in for certain missing elements. For example, if you throw in \sqrt{-5} then we have 9 = 3\times 3 = \bigl(2+\sqrt{-5}\bigr)\bigl(2-\sqrt{-5}\bigr), (in particular, 3 is no longer prime), but in terms of ideals (9)=\bigl(3,2+ \sqrt{-5}\bigr)^2\bigl(3,2- \sqrt{-5}\bigr)^2 is a unique factorization in terms of prime ideals. Note, e.g., (3)=(3,2+ \sqrt{-5})(3,2-\sqrt{-5}) but those prime ideals do not correspond to single elements. ---
**Author:** ![The\_Other\_Waldo\_Pepper](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_other_waldo_pepper/32/12370_2.png) [@The\_Other\_Waldo\_Pepper](https://boards.straightdope.com/u/The_Other_Waldo_Pepper)\ **Post date:** [April 1, 2025, 11:04am UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/50 "2025-04-01T11:04:15Z")
> [@Pleonast](#): > > 2: di-visible > 3: tri-visible > 4: quad-visible > 5: pen-visible > etc… _“Today in school we learned all about stuff that’s sex-visible.”_ ---
**Author:** ![Ponderoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ponderoid/32/19221_2.png) [@Ponderoid](https://boards.straightdope.com/u/Ponderoid)\ **Post date:** [April 1, 2025, 11:26am UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/51 "2025-04-01T11:26:57Z")
> [@DPRK](#): > > Any word for “divisible by 10”? Kinda. “round” But it’s not that precise. It means divisible by the nearest power of 10 in the current context. [Previous page](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346.md?page=2)