# 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:** 20\ **Page:** 1
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [March 30, 2025, 3:55pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/1 "2025-03-30T15:55:11Z")
My theory is that it’s just a matter of convenience. It’s easier to say “for every prime number” rather than always having to say “for every prime number except one.” Anyone got a better reason? ---
**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:** [March 30, 2025, 3:58pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/2 "2025-03-30T15:58:56Z")
The most important thing about primes is what’s called the unique prime factorization theorem. Given any positive integer, there is one and only one to express it as a product of primes. So, for instance, 12 = 2\*2\*3, and 42 = 2\*3\*7. But if we counted 1 as a prime, then there’d be an infinite number of ways, since 12 is also 1\*2\*2\*3, or 1\*1\*1\*1\*2\*2\*3, etc. ---
**Author:** ![Northern\_Piper](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/northern_piper/32/5304_2.png) [@Northern\_Piper](https://boards.straightdope.com/u/Northern_Piper)\ **Post date:** [March 30, 2025, 4:11pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/3 "2025-03-30T16:11:32Z")
Chronos, that looks interesting, but I’m afraid my math-deficient brain isn’t scanning it. Could you expand a bit for those of us for whom math isn’t intuitive? Why is 12 = 223? ---
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [March 30, 2025, 4:12pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/4 "2025-03-30T16:12:36Z")
> [@Chronos](#): > > Given any positive integer, there is one and only one to express it as a product of primes. Couldn’t you just say, “Given any positive integer there is one and only one way to express it as a product of primes other than one.”? ---
**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 30, 2025, 4:18pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/5 "2025-03-30T16:18:07Z")
I’m sure there’s some munched multiplies in there. > So, for instance, 12 = 2\*2\*3, and 42 = 2\*3\*7. But if we counted 1 as a prime, then there’d be an infinite number of ways, since 12 is also 1\*2\*2\*3, or 1\*1\*1\*1\*2\*2\*3, etc. ---
**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 30, 2025, 4:31pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/6 "2025-03-30T16:31:29Z")
Yes, we could, but it’s longer and harder to deal with than a simple rule without an exception. Ultimately it _is_ a matter of convenience: a lot of theorems and things would get more complicated if 1 were considered a prime, and very few would get simpler. ---
**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:** [March 30, 2025, 4:33pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/7 "2025-03-30T16:33:16Z")
> [@Chronos](#): > > Given any positive integer, there is one and only one to express it as a product of primes. Is 1 a positive integer? If so, how do you express it as a product of primes? ---
**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:** [March 30, 2025, 4:34pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/8 "2025-03-30T16:34:44Z")
Gah, I forgot that asterisks make Discourse go stupid. I’ve fixed my post. > [@Ynnad](#): > > Couldn’t you just say, “Given any positive integer there is one and only one way to express it as a product of primes other than one.”? Yes, but since statements like that are the reason we’re interested in primes in the first place, why ever include 1 at all? > [@The\_Other\_Waldo\_Pepper](#): > > Is 1 a positive integer? If so, how do you express it as a product of primes? 1 is the product of no primes. No primes is the only way to express 1, and it’s the only number that’s the product of no primes. ---
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [March 30, 2025, 4:49pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/9 "2025-03-30T16:49:12Z")
> [@Chronos](#): > > … [W]hy ever include 1 at all? Maybe so that you could give third-graders (or maybe the average American) a simple definition of prime numbers rather than having to explain the exception for one over and over again. ---
**Author:** ![leahcim](https://avatars.discourse-cdn.com/v4/letter/l/b4bc9f/32.png) [@leahcim](https://boards.straightdope.com/u/leahcim)\ **Post date:** [March 30, 2025, 4:59pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/10 "2025-03-30T16:59:34Z")
> [@Ynnad](#): > > primes other than one. This is what it comes down to. “The primes” as they are currently defined now are a useful set the shows up in a lot of theorems. “The primes and also one” is a less useful set that doesn’t show up in very many theorems. If you redefine the primes to include one, you end up having to say “primes other than one” a _lot_ more, and don’t get very much savings from being able to say just “the primes” for those few theorems where “primes + 1” is an important set. ---
**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:** [March 30, 2025, 5:00pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/11 "2025-03-30T17:00:30Z")
> [@Ynnad](#): > > Maybe so that you could give third-graders (or maybe the average American) a simple definition of prime numbers rather than having to explain the exception for one over and over again. “A number that has exactly two factors, 1 and itself” ---
**Author:** ![hogarth](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hogarth/32/1773_2.png) [@hogarth](https://boards.straightdope.com/u/hogarth)\ **Post date:** [March 30, 2025, 5:03pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/12 "2025-03-30T17:03:18Z")
> [@Chronos](#): > > A number that has exactly two factors, 1 and itself Like -1 = -1 x 1? 😛 ---
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [March 30, 2025, 5:04pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/13 "2025-03-30T17:04:29Z")
> [@Chronos](#): > > “A number that has exactly two factors, 1 and itself” 1 x 1 = 1 How many factors are in that equation? ---
**Author:** ![Zakalwe](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/zakalwe/32/270_2.png) [@Zakalwe](https://boards.straightdope.com/u/Zakalwe)\ **Post date:** [March 30, 2025, 5:12pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/14 "2025-03-30T17:12:58Z")
> [@Chronos](#): > > “A number that has exactly two factors, 1 and itself” No. It has infinite factors: 1 x 1 1 x 1 x 1 1 x 1 x 1 ad infinitum Note that if 1 is prime this is true of everything: 2 x 3 x 1 x 1 x 1 x 1 is still 6. > [@Chronos](#): > > Given any positive integer, there is one and only one to express it as a product of primes. Surely you meant for any _non-prime_ positive integer…5 is a positive integer, but is not factorable as the product of 2 primes - because it is prime. ---
**Author:** ![Topologist](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/topologist/32/3208_2.png) [@Topologist](https://boards.straightdope.com/u/Topologist)\ **Post date:** [March 30, 2025, 5:19pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/15 "2025-03-30T17:19:01Z")
> [@Zakalwe](#): > > > [@Chronos](#): > > > > Given any positive integer, there is one and only one to express it as a product of primes. > > Surely you meant for any _non-prime_ positive integer…5 is a positive integer, but is not factorable as the product of 2 primes - because it is prime. When a mathematician refers to a product, it could be the product of as few as 0 factors, the product of no factors being 1. (Think of it as starting with 1 and then multiplying by all of the given factors. If there are none, you just have 1 as the result.) If you have just one factor, that factor is your result. So, prime numbers also have unique expressions as products of primes, with just one factor, being the prime itself. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [March 30, 2025, 9:38pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/16 "2025-03-30T21:38:15Z")
Yeah. It’s bad enough having to count 2 as a prime - there’s a number of theorems about “every odd prime” (meaning every prime but 2). Adding 1 to the club would (as @Chronos and others have explained) much worse ---
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [March 30, 2025, 10:01pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/17 "2025-03-30T22:01:41Z")
> [@leahcim](#): > > If you redefine the primes to include one, you end up having to say “primes other than one” a _lot_ more, and don’t get very much savings from being able to say just “the primes” … . So you agree that it’s that way just because it’s more convenient. Not that there is anything wrong with that. ---
**Author:** ![Zoobi](https://avatars.discourse-cdn.com/v4/letter/z/13edae/32.png) [@Zoobi](https://boards.straightdope.com/u/Zoobi)\ **Post date:** [March 30, 2025, 10:07pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/18 "2025-03-30T22:07:28Z")
It also makes functions like > **[Euler's totient function](https://en.wikipedia.org/wiki/Euler%27s_totient_function)** > > In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as > > > > φ > ( > n > ) > > > {\\displaystyle \\varphi (n)} > > or > > > > ϕ > ( > n > ) > > > {\\displaystyle \\phi (n)} > > , and may also be called Euler's phi function. In other words, it is the number of integers k in the range 1 ≤ k ≤ n for w... work. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [March 30, 2025, 10:52pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/19 "2025-03-30T22:52:35Z")
I’m still not understanding who is inconvenienced by the definition. It’s a technical term; a technical definition is to be expected. Should math also drop the word exponent because it’s confusing? It also has a _non_-mathematical definition. > a person who believes in and promotes the truth or benefits of an idea or theory. > > “an early **exponent of** the teachings of Thomas Aquinas” ---
**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:** [March 30, 2025, 10:54pm UTC](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346/20 "2025-03-30T22:54:55Z")
> [@Ynnad](#): > > So you agree that it’s that way just because it’s more convenient. Well, yes, every definition in math is because it’s convenient. We could do without definitions, and just use the full description instead of the single word every time, but that would be really inconvenient. > [@Andy\_L](#): > > Yeah. It’s bad enough having to count 2 as a prime - there’s a number of theorems about “every odd prime” (meaning every prime but 2). There are some theorems about the odd primes, but the ones for which you do want to count 2 are far more numerous and significant. And for that matter, there are some for which you want to exclude both 2 and 3, like “(almost) all primes are of the form 6n+1 or 6n+5”. Or exclude 2, 3, and 5, or whatever. [Next page](https://boards.straightdope.com/t/why-isnt-1-a-prime-number/1016346.md?page=2)