# Why is 1 not a prime number? **URL:** \ **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:** 20\ **Page:** 2
**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 16, 2019, 12:04am UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/22 "2019-04-16T00:04:45Z")
That only loses you prime factorization if 1+sqrt(-5) is a prime. But it divides 2\*3, even though it doesn’t divide 2 or 3, so it’s composite, and (1+sqrt(-5))(1-sqrt(-5)) is not a prime factorization. Although… then I suppose you can ask what the prime factorization of sqrt(-5) is. But on the gripping hand, just adjoining a single complex irrational to the integers breaks all sorts of properties, like closure. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [April 16, 2019, 1:38am UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/23 "2019-04-16T01:38:59Z")
> [@Chronos](#): > > That only loses you prime factorization if 1+sqrt(-5) is a prime. But it divides 2\*3, even though it doesn’t divide 2 or 3, so it’s composite, and (1+sqrt(-5))(1-sqrt(-5)) is not a prime factorization. > > Although… then I suppose you can ask what the prime factorization of sqrt(-5) is. > > But on the gripping hand, just adjoining a single complex irrational to the integers breaks all sorts of properties, like closure. Sorry, you are right. Adjoining sqrt(-5) breaks prime factorization because, as the example shows, not every number factors into primes in the first place. Once every element factors into primes, you have uniqueness in the usual way. Another way to look at the example would be that the fact that there are two essentially different factorizations into irreducible elements shows that the ring in question fails to be a unique factorization domain. What property of “closure” did you mean is broken, though? ETA isn’t sqrt(-5) itself already prime? ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [April 16, 2019, 3:20am UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/24 "2019-04-16T03:20:44Z")
I’ve not ever seen a definition of prime numbers that allows for negative numbers. They all say “an integer [or whole number] greater than 1…” or “can only be evenly divided by itself and 1.” Can anyone show me a cite for the idea that negative numbers can also be considered prime? ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [April 16, 2019, 3:36am UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/25 "2019-04-16T03:36:00Z")
> [@BigT](#): > > I’ve not ever seen a definition of prime numbers that allows for negative numbers. They all say “an integer [or whole number] greater than 1…” or “can only be evenly divided by itself and 1.” > > Can anyone show me a cite for the idea that negative numbers can also be considered prime? [_Algebraic Number Theory_](http://www.jmilne.org/math/CourseNotes/ANTe6.pdf), by J. S. Milne, Introduction, page 1: > [@](#): > > The _fundamental theorem of arithmetic_ says that every nonzero integer _m_ can be written in the form, > > m = ± p[sub]1[/sub]…p[sub]n[/sub],   p[sub]i[/sub] a prime number, > > and that this factorization is essentially unique. > > Consider more generally an integral domain A. An element a ∈ A is said to be a _unit_ if it has an inverse in A (element b such that ab = 1 = ba). I write A[sup]×[/sup] for the multiplicative group of units in A. An element π of A is said to [be] _prime_ if it is neither zero nor a unit, and if > > π | ab ⇒ π | a or π | b . ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [April 16, 2019, 3:49am UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/26 "2019-04-16T03:49:46Z")
NB I see that he explains more clearly what I said in my reply to **Chronos** : “Why does unique factorization fail [in **Z** [√-5]]? The problem is that irreducible elements need not be prime. In the above example, 1 + √-5 divides 2⋅3 but it divides neither 2 nor 3. In fact, in an integral domain in which factorizations exist… factorization is unique if all irreducible elements are prime.” ---
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [April 16, 2019, 1:09pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/27 "2019-04-16T13:09:55Z")
> [@DPRK](#): > > [_Algebraic Number Theory_](http://www.jmilne.org/math/CourseNotes/ANTe6.pdf), by J. S. Milne, Introduction, page 1: That just shows how you can define a prime factorization of a negative number though, and elsewhere in the same work he writes: > [@](#): > > Throughout the notes, p is a prime number, i.e., > _p_ = 2, 3, 5, … Not saying you’re wrong, as I’m out of my depth here. But that specific quote, and the notes on basic definitions used in the work, seem to stick with only positive primes. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [April 16, 2019, 1:41pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/28 "2019-04-16T13:41:16Z")
> [@naita](#): > > That just shows how you can define a prime factorization of a negative number though It looks that way to me, too. The ± allows the m to be negative even though the p’s are not. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [April 16, 2019, 2:18pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/29 "2019-04-16T14:18:10Z")
I quoted some context, but the last sentence defines what it means for an element of an integral domain to be “prime”, namely that if p divides ab then it divides one of the factors a or b. This is a generalization of what it means for a positive rational integer to be prime (the rational integers being the prototypical example of an integral domain). In particular, negative integers are “prime” under this definition, and multiplying a “prime” by a unit leaves it prime. Let us ask, can anyone cite a book or article wherein negative numbers are _not_ prime, as soon as negative integers are considered, rather than only looking at natural numbers? Because the text I linked to is not the only one where negative numbers may be prime. ---
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [April 16, 2019, 2:51pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/30 "2019-04-16T14:51:41Z")
> [@DPRK](#): > > I quoted some context, but the last sentence defines what it means for an element of an integral domain to be “prime”, namely that if p divides ab then it divides one of the factors a or b. This is a generalization of what it means for a positive rational integer to be prime (the rational integers being the prototypical example of an integral domain). In particular, negative integers are “prime” under this definition, and multiplying a “prime” by a unit leaves it prime. > > Let us ask, can anyone cite a book or article wherein negative numbers are _not_ prime, as soon as negative integers are considered, rather than only looking at natural numbers? Because the text I linked to is not the only one where negative numbers may be prime. The book you quote states : “Throughout the notes, p is a prime number, i.e., p = 2, 3, 5, …” In a very practical sense that means that -2 might be “prime”, but it’s the prime 2 with a negative unit. ---
**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 16, 2019, 7:16pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/31 "2019-04-16T19:16:46Z")
Nobody talks much about the primes in **Z** , because the negative primes are just the familiar primes in **N** with a negative sign, so there’s nothing new to be said. On the other hand, the primes in the complex integers are different, as evidenced by 5 not being prime in that domain, so they get attention again. ---
**Author:** ![KarlGauss](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/karlgauss/32/3713_2.png) [@KarlGauss](https://boards.straightdope.com/u/KarlGauss)\ **Post date:** [April 16, 2019, 9:23pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/32 "2019-04-16T21:23:05Z")
Disclaimer: I really don’t know anything about number theory, especially analytic number theory. Still, may I ask: do primes in other domains also link to the Riemann Hypothesis? Or are there separate analogues? Neither? ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [April 17, 2019, 6:45pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/33 "2019-04-17T18:45:05Z")
> [@KarlGauss](#): > > Disclaimer: I really don’t know anything about number theory, especially analytic number theory. > > Still, may I ask: do primes in other domains also link to the Riemann Hypothesis? Or are there separate analogues? Neither? I read this as a two-part question. One can define an analogue of the classical Riemann zeta function for any algebraic number field; this was done by Dedekind in 1863, and today one speaks of Dedekind zeta-functions. (And one can obtain results on the distribution of primes in number fields.) This naturally leads to an “extended” Riemann hypothesis which would apply to any number field. (And of course such zeta functions may be and are generalized even further.) Note that the various generalized Riemann hypotheses do include as a particular case the classical Riemann hypothesis. Now, the first part of your question seems to be whether the _ordinary_ RH can be reformulated as an equivalent problem involving the distribution of primes in a more general setting, more generalized zeta functions or some other arithmetic or analytic problem. Certainly mathematicians have endeavored to establish such bridges, because you need non-elementary tools with which to attack the problem, but I don’t necessarily feel qualified to survey all the precise statements. It is worth pointing out that today no one yet knows a proof of the RH so it’s not like there is one obvious thing to try. ---
**Author:** ![GuyTanzer](https://avatars.discourse-cdn.com/v4/letter/g/48db29/32.png) [@GuyTanzer](https://boards.straightdope.com/u/GuyTanzer)\ **Post date:** [April 17, 2019, 11:22pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/34 "2019-04-17T23:22:14Z")
All of which indeed proves, once and for all: One _is_ the loneliest number. QED. ---
**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 17, 2019, 11:26pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/35 "2019-04-17T23:26:12Z")
Nah, zero is the loneliest number. 1 has -1, i, and -i to keep it company in the Units Club, but there’s only one zero. ---
**Author:** ![KarlGauss](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/karlgauss/32/3713_2.png) [@KarlGauss](https://boards.straightdope.com/u/KarlGauss)\ **Post date:** [April 17, 2019, 11:36pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/36 "2019-04-17T23:36:11Z")
> [@DPRK](#): > > I read this as a two-part question. > > One can define an analogue of the classical Riemann zeta function for any algebraic number field; this was done by Dedekind in 1863, and today one speaks of Dedekind zeta-functions. (And one can obtain results on the distribution of primes in number fields.) This naturally leads to an “extended” Riemann hypothesis which would apply to any number field. (And of course such zeta functions may be and are generalized even further.) Note that the various generalized Riemann hypotheses do include as a particular case the classical Riemann hypothesis. > > Now, the first part of your question seems to be whether the _ordinary_ RH can be reformulated as an equivalent problem involving the distribution of primes in a more general setting, more generalized zeta functions or some other arithmetic or analytic problem. Certainly mathematicians have endeavored to establish such bridges, because you need non-elementary tools with which to attack the problem, but I don’t necessarily feel qualified to survey all the precise statements. It is worth pointing out that today no one yet knows a proof of the RH so it’s not like there is one obvious thing to try. Thank you for answer. Very helpful. What you note is very much the type of thing I was wondering about. ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [April 18, 2019, 12:04pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/37 "2019-04-18T12:04:55Z")
> [@Chronos](#): > > Nah, zero is the loneliest number. 1 has -1, i, and -i to keep it company in the Units Club, but there’s only one zero. Not in two’s complement binary. 00…0000 and 10…000 are both zeros. The first is positive zero and the second is negative zero. They both have the property that if you negate them in th usual way (flip all bits and add one) you get the same number back. So, the negation of positive zero is positive zero and the negation of negative zero is negative zero. Somehow that sentence makes sense to us computer folk. (Note that since “-0” isn’t often be used in practice, some people _define_ it as the negative of 2 raised to the (word size - 1) in order to “squeeze out” an extra value. This is an ugly, ugly kludge that breaks a lot of binary arithmetic and if you need that extra value you should be using a larger word size.) ---
**Author:** ![Dead\_Cat](https://avatars.discourse-cdn.com/v4/letter/d/9fc29f/32.png) [@Dead\_Cat](https://boards.straightdope.com/u/Dead_Cat)\ **Post date:** [April 18, 2019, 12:59pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/38 "2019-04-18T12:59:30Z")
> [@Indistinguishable](#): > > The parenthetical exclusion of units will seem ad hoc, but should be unified with the rest of the definition: > > A prime number, in this sense, is one which only divides the product of some list of values when there is some value in that list which it divides. This is true not just of lists of 2 values, but also lists of 3 values, lists of 4 values… and even lists of 0 values. > > A prime only divides the product of an empty list (which comes out to 1) when there is some value in that list which it divides (which of course can’t happen, since the list is empty); thus, a prime is not allowed to divide 1. > > So the exclusion of 1 and other units from the prime numbers in this sense is hardly ad hoc; it’s just part of the same divisibility condition. (The exclusion of zero from the primes is genuinely a separate condition here, though; it would otherwise be a, well, prime example of such a thing.) Quoting this post just to say “welcome back” (I haven’t seen you post in a while, though that could just be me) and to note that rarest of beasts - an unedited post by Indistinguishable :). ---
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [April 18, 2019, 3:29pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/39 "2019-04-18T15:29:48Z")
> [@ftg](#): > > Not in two’s complement binary. 00…0000 and 10…000 are both zeros. The first is positive zero and the second is negative zero. They both have the property that if you negate them in th usual way (flip all bits and add one) you get the same number back. That’s just a representation of numbers though, we can introduce -0 in all the other representations as well, it just wouldn’t give us anything useful. ---
**Author:** ![Triskadecamus](https://avatars.discourse-cdn.com/v4/letter/t/b19c9b/32.png) [@Triskadecamus](https://boards.straightdope.com/u/Triskadecamus)\ **Post date:** [April 18, 2019, 3:38pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/40 "2019-04-18T15:38:21Z")
> [@naita](#): > > That’s just a representation of numbers though, we can introduce -0 in all the other representations as well, it just wouldn’t give us anything useful. But it would encourage some people to ask about +0, and then there would have to be a whole argument on the subtle implications their differences. Mathematicians don’t need any more toys. Tris * * * Think of a number. Now try not to. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [April 18, 2019, 4:06pm UTC](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549/41 "2019-04-18T16:06:06Z")
In mathematics, +0 = -0. The computer engineer’s nightmare therefore begins as soon as these have different or multiple machine-level representations, because the hardware will have to check for it every time there is an equality test or comparison, and who knows what bugs could still ensue. Back to mathematics, in a ring (i.e. algebraic structure in which you have addition and multiplication) you always have to have zero, but it is conceivable that 1 = 0, in which case zero could claim that I am the Alpha and the Omega, the First and the Last, the Beginning and the End. [Previous page](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549.md?page=1) [Next page](https://boards.straightdope.com/t/why-is-1-not-a-prime-number/832549.md?page=3)