[HN Gopher] Is 1 Prime, and Does It Matter?
___________________________________________________________________
Is 1 Prime, and Does It Matter?
Author : jamespropp
Score : 57 points
Date : 2025-04-21 19:25 UTC (3 hours ago)
(HTM) web link (mathenchant.wordpress.com)
(TXT) w3m dump (mathenchant.wordpress.com)
| JJMcJ wrote:
| One reason that 1 is often excluded from the prime numbers is
| that if it was included, it would complicate the theorems,
| proofs, and exposition by the endless repetition of "not equal to
| 1".
| reaperman wrote:
| Yes, it's more of a convention where we assume language like
| "...ignoring the trivial case of 1 being an obvious factor of
| every integer." It's not interesting or meaningful, so we
| ignore it for most cases.
| gerdesj wrote:
| I'm no expert but:
|
| "...ignoring the trivial case of 1 being an obvious factor of
| every integer."
|
| I remember quite a big chunk of GEB formally defining how
| integers are really not trivial! The main problem seems to be
| is that you soon end up with circular reasoning if you are
| not razor sharp with your definitions. That's just in an
| explainer book 8)
|
| Then you have to define what factor means ...
| Maxatar wrote:
| Correct, it's impossible to specifically and formally
| define the natural numbers so that addition and
| multiplication work. Any definition of the natural numbers
| will also define things that look very similar to natural
| numbers but are not actually natural numbers.
| stouset wrote:
| Do you have a link to where I could learn more about
| this?
| gerdesj wrote:
| You might start here:
| https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach
|
| That's the GEB I mentioned above.
| pinkmuffinere wrote:
| I know this is a great book, it's been on my to-read list
| for about 5 years. But I never get to it. Is there not
| another (shorter) discussion I could read on this? Even
| an academic paper would be acceptable.
| gerdesj wrote:
| As I said above, I'm not an expert. However, I read GEB
| on a whim when bored at school and I think it still
| informs my thinking 35 years later.
|
| Move GEB up the reading list right now! The edition I
| initially read was hard bound and was quite worn. I
| bought and read it again about 20 years ago and found
| more treasures.
|
| It is a proper nerd grade treatise for non experts who
| are interested in maths, music and art. Really: maths,
| music and art from a mostly mathematical perspective.
| Hofstadter's writing style is very easy going and he is a
| master of clarity without complexity.
|
| I don't think you need any more Maths than you would get
| up to age 18 or so at school to understand the entire
| book and probably less. Even if you gloss the formal
| Maths the book still works.
| Maxatar wrote:
| There's no shortage of blog posts on the topic, but here
| is one that is fairly rigorous but doesn't assume too
| much background knowledge:
|
| https://risingentropy.com/a-result-on-the-incompleteness-
| of-...
| overboard2 wrote:
| What do you mean by "not actually"?
|
| Edit: do you mean literally impossible?
| Maxatar wrote:
| I mean it's logically impossible to formally and
| specifically define the natural numbers without
| introducing a logical inconsistency. The best you can do
| is define a set that has all the properties of natural
| numbers but will also define things that aren't natural
| numbers as well.
|
| As an analogy you could imagine trying to define the set
| of all animals with a bunch of rules... "1. Animals have
| DNA, 2. Animals ingest organic matter. 3. Animals have a
| nervous system. 4. ... etc..."
|
| And this is true of all animals, but it will also be true
| of things that aren't animals as well, like slime molds
| which are not quite animals but very similar to them.
|
| Okay so you keep adding more rules to narrow down your
| definition and stamp out slime molds, but you find some
| other thing satisfy that definition...
|
| Now for animals maybe you can eventually have some very
| complex rule set that defines animals exactly and rules
| out all non-animals, but the principle is that this is
| not possible for natural numbers.
|
| We can have rules like "0" is a natural number. For every
| natural number N there is a successor to it N + 1. If N +
| 1 = M + 1 then N = M. There is no natural number Q such
| that Q + 1 = 0.
|
| Okay this is a good starting point... but just like with
| animals there are numbers that satisfy all of these rules
| but aren't natural numbers. You can keep adding more and
| more rules to try to stamp these numbers out, but no
| matter how hard, even if you add infinitely many rules,
| there will always be infinitely many numbers that satisfy
| your rules but aren't natural numbers.
|
| In particular what you really want to say is that a
| natural number is finite, but no matter how hard you try
| there is no formal way to actually capture the concept of
| what it means to be finite in general so you end up with
| these mutant numbers that satisfy all of your rules but
| have infinitely many digits, and these are called non-
| standard natural numbers.
|
| The reason non-standard natural numbers are a problem is
| because you might have a statement like "Every even
| integer greater than 2 can be written as the sum of two
| primes." and this statement might be true of the actual
| natural numbers but there might exist some freak mutant
| non-standard natural number for which it's not true.
| Unless your rules are able to stamp out these mutant non-
| standard natural numbers, then it is not possible to
| prove this statement, the statement becomes undecidable
| with respect to your rules. The only statements you can
| prove with respect to your rules are statements that are
| true of the real natural numbers as well as true of all
| the mutant natural numbers that your rules have not been
| able to stamp out.
|
| So it's in this sense that I mean that it's not possible
| to specifically define the natural numbers. Any
| definition you come up with will also apply to mutant
| numbers, and these mutant numbers can get in the way of
| you proving things that are in principle true about the
| actual natural numbers.
| gerdesj wrote:
| It seems you know what you are on about! Thank you for a
| cracking comment.
|
| I've always had this feeling that the foundations
| (integers etc) are a bit dodgy in formal Maths but just
| as with say Civil Engineering, your world hasn't fallen
| apart for at least some days and it works. Famously, in
| Physics involving quantum: "Shut up and calculate".
|
| Thankfully, in the real world I just have to make web
| pages, file shares and glittery unicorns available to the
| computers belonging to paying customers. Securely ...
|
| The foundational aspect equivalent of integers in IT
| might be DNS. Fuck around with either and you come
| unstuck rather quickly without realising exactly why
| until you get suitably rigorous ...
|
| I'm also a networking bod (with some jolly expensive test
| gear) but that might be compared to pencils and paper for
| Maths 8)
| pja wrote:
| > Correct, it's impossible to specifically and formally
| define the natural numbers so that addition and
| multiplication work. Any definition of the natural
| numbers will also define things that look very similar to
| natural numbers but are not actually natural numbers.
|
| Are such objects not inevitably isomorphic to the natural
| numbers?
|
| Can you give an example of a formal definition that leads
| to something that obviously isn't the same as the
| naturals?
| btilly wrote:
| The Peano Axioms lead to both the standard model of
| arithmetic (the integers that we want), and nonstandard
| models. See https://en.wikipedia.org/wiki/Non-
| standard_model_of_arithmet....
|
| In that article you'll see references to "first order
| logic" and "second order logic". First order logic
| captures any possible finite chain of reasoning. Second
| order logic allows us to take logical steps that would
| require a potentially infinite amount of reasoning to do.
| Godel's famous theorems were about the limitations of
| first order logic. While second order logic has no such
| limitations, it is also not something that humans can
| actually do. (We can reason about second order logic
| though.)
|
| Anyways a nonstandard model of arithmetic can have all
| sorts of bizarre things. Such as a proof that Peano
| Axioms lead to a contradiction. While it might seem that
| this leads to a contradiction in the Peano Axioms, it
| doesn't because the "proof" is (from our point of view)
| infinitely long, and so not really a proof at all! (This
| is also why logicians have to draw a very careful
| distinction between "these axioms prove" and "these
| axioms prove that they prove"...)
| john-h-k wrote:
| Can you elaborate on this?
|
| My understanding is you can specifically and formally
| define the natural numbers with addition and
| multiplication, although multiplication means the
| language is no longer decidable.
|
| You can define natural numbers with just addition (
| Presburger arithmetic ) and it's decidable.
|
| Im not sure how undecidable <=> "will define things that
| are similar to natural numbers but are not" but maybe I
| am missing something
| Maxatar wrote:
| Yeah for sure.
|
| If a sentence S is undecidable from your axioms for the
| natural numbers then there are two models A and B
| satisfying those axioms where A satisfies S and B
| satisfies not S. So which one is the standard natural
| numbers, is it A or B?
|
| Either A or B will be an example of something that
| satisfies your definition of natural numbers and yet is
| not the natural numbers.
| sam_ezeh wrote:
| >Any definition of the natural numbers will also define
| things that look very similar to natural numbers but are
| not actually natural numbers
|
| This isn't correct. This is only true for first-order
| theories of the natural numbers using the axiom schema of
| induction. Second-order Peano arithmetic with the full
| axiom of induction has the natural numbers as its only
| model. This property is called "categoricity" and you can
| find the proof here [1] if you're interested
|
| [1]: https://builds.openlogicproject.org/content/second-
| order-log...
| JadeNB wrote:
| > One reason that 1 is often excluded from the prime numbers is
| that if it was included, it would complicate the theorems,
| proofs, and exposition by the endless repetition of "not equal
| to 1".
|
| This is true and compelling as things developed, but I think
| it's an explanation of where history brought us, rather than a
| logical inevitability. For example, I can easily imagine, in a
| different universe, teachers patiently explaining that we
| declare that the empty set is not a set, to avoid complicating
| theorems, proofs, and exposition by the endless repetition of
| "non-empty set."
|
| (I agree that this is different, because there's no interesting
| "unique factorization theorem" for sets, but I can still
| imagine things developing this way. And, indeed, there _are_
| complications caused by allowing the empty set in a model of a
| structure, and someone determined to do so can make themselves
| pointlessly unpopular by asking "but have you considered the
| empty manifold?" and similar questions. See also
| https://mathoverflow.net/questions/45951/interesting-
| example....)
| tux3 wrote:
| That's an interesting thought, but I think that'd break the
| usual trick of building up objects from the empty set, a set
| containing the empty set, then the set containing both of
| those and so forth.
|
| That universe would be deprived from the bottomless
| wellspring of dryness that is the set theoretic foundations
| of mathematics. Unthinkable!
| JadeNB wrote:
| > That universe would be deprived from the bottomless
| wellspring of dryness that is the set theoretic foundations
| of mathematics. Unthinkable!
|
| "Wellspring of dryness" is quite a metaphor, and I take it
| from that metaphor that this outcome wouldn't much bother
| you. I'll put in a personal defense for set theory, but
| only an appeal to my personal taste, since I have no
| expert, and barely even an amateurish, knowledge of set
| theory beyond the elementary; but I'll also acknowledge
| that set-theoretic foundations are not to everyone's taste,
| and that someone who has an alternate foundational system
| that appeals to them is doing no harm to themselves or to
| me.
|
| > That's an interesting thought, but I think that'd break
| the usual trick of building up objects from the empty set,
| a set containing the empty set, then the set containing
| both of those and so forth.
|
| In this alternate universe, the ZF or ZFC axioms (where C
| becomes, of course, "the product of sets is a set") would
| certainly involve, not the axiom of the empty set, but
| rather some sort of "axioms of sets", declaring that there
| exists a set. Because it's not empty, this set has at least
| one element, which we may extract and use to make a one-
| element set. Now observe that all one-element sets are set-
| theoretically the same, and charge ahead with the
| construction not O, {O}, {O, {O}}, etc. but *, * [?] {*}, *
| [?] {*} [?] {* [?] {*}}, etc. Then all that would be left
| would be to decide whether our natural numbers started at
| the cardinality 1 of *, or if we wanted natural numbers to
| count quantities 1 less than the cardinality of a set.
| gus_massa wrote:
| Many (most?) results are easier to write if you allow the
| empty set. For example:
|
| " _The intersection of two sets is a set._ "
| JadeNB wrote:
| > Many (most?) results are easier to write if you allow the
| empty set. For example:
|
| > "The intersection of two sets is a set."
|
| Many results in set theory, yes! (Or at least in elementary
| set theory. I'm not a set theorist by profession, so I
| can't speak to how often it arises in research-level set
| theory.) But, once one leaves set theory, the empty set can
| cause problems. For the first example that springs to mind,
| it is a cute result that, if a set S has a binary operation
| * such that, for every pair of elements a, b in S, there is
| a unique solution x to a*x = b, and a unique solution y to
| y*a = b, then * makes S a group ... unless S is empty!
|
| In fact, on second thought, even in set theory, there are
| things like: the definition of a partial order being a well
| ordering would become simpler to state if the empty set
| were disallowed; and the axiom of choice would become just
| the statement that the product of sets is a set! I'm sure
| that I could come up with more examples where allowing
| empty sets complicates things, just as you could come up
| with more examples where it simplifies them. That there is
| no unambiguous answer one direction or the other is why I
| believe this alternate universe could exist, but we're not
| in it!
| murderfs wrote:
| A good example of this is the natural numbers. Algebraists
| usually consider zero to be a natural number because
| otherwise, it's not a monoid and set theorists want zero
| because it's the size of the empty set. My number theory
| textbook defined natural numbers as positive integers, but
| I'm not entirely sure why.
| mathgeek wrote:
| > My number theory textbook defined natural numbers as
| positive integers, but I'm not entirely sure why.
|
| Since both the inclusion and exclusion of zero are accepted
| definitions depending on who's asking, books usually just
| pick one or define two sets (commonly denoted as N_0 and
| N_1). Different topics benefit from using one set over the
| other, as well as having to deal with division by zero,
| etc. Number theory tends to exclude zero.
| tikhonj wrote:
| And the reason we'd have to constantly exclude 1 is that it
| behaves in a qualitatively different way than prime numbers--
| and understand what this means and why that's the case is the
| real insight here.
| jordigh wrote:
| To be fair, 2 is also a very odd prime because it's even.
|
| So many theorems have to say, "for every odd prime..."
|
| https://math.stackexchange.com/questions/1177104/what-is-an-...
| kordlessagain wrote:
| The concept of "one" holds a dual role. It represents a
| countable unit: something you can put in a bowl and also
| stands for indivisibility itself. When you divide any
| quantity by an indivisible unit, you're simply counting how
| many of those indivisibles fit within it. Then comes 2: the
| first number that is divisible, but only by itself and the
| indivisible one. That's what makes it prime. A prime is a
| number divisible only by itself and by 1, the indivisible
| origin of all counting.
| aleph_minus_one wrote:
| > Then comes 2: the first number that is divisible, but
| only by itself and the indivisible one.
|
| This does hold in the ring Z. In the ring Z[i], 2 =
| (1+i)*(1-i), and the two factors are prime elements.
| brennopost wrote:
| It's actually the least odd prime
| chrismcb wrote:
| It isn't odd at all! And that I'm being pendantic. But you
| can't say it is very odd, and then I'm the next sentence day
| "for every odd prime..."
| wesselbindt wrote:
| If 1 is prime, then the fundamental theorem of arithmetic goes
| from "every positive integer can be written as a product* of
| primes in one and only one way" to "every positive integer can be
| written as a product of primes greater than 1 in one and only one
| way". Doesn't quite have the same ring to it. So just from an
| aesthetic perspective, no I'd rather 1 isn't a prime number.
|
| * empty products being 1 of course
| apetresc wrote:
| Not just that one; practically every useful theorem about
| primes would have to be rewritten to "if p is a prime other
| than 1".
| SketchySeaBeast wrote:
| Isn't "every positive integer can be written as a product of
| primes greater than 1 in one and only one way" incorrect? A
| prime number is a only product of itself * 1, isn't it?
| wesselbindt wrote:
| 1 is not greater than 1, and a product of one prime is still
| a product of primes
| SketchySeaBeast wrote:
| Yeah, I didn't understand you can have a product of a
| single number.
| jdoliner wrote:
| Mathematicians generally feel that a single number qualifies
| as a "product of 1 number." So 7 can be written as just 7
| which is still considered a product of prime(s). This is
| purely a convention thing to make it so theorems can be
| stated more succinctly, as with not counting 1 as prime.
| SketchySeaBeast wrote:
| Ah, OK, thank you.
| laweijfmvo wrote:
| i remember something from math class about "1" and "prime"
| being special cases of "units" and "irreducible" (?) that made
| me think these kinds of definitions are much more complicated
| than we want them to be regardless.
| wesselbindt wrote:
| The first part of your comment is completely correct. The
| latter is a matter of taste, of course. I think the main
| thing that can be said for a lot of the definitions we have
| in algebra is that the ones we're using are the ones that
| stood the test of time because they turned out to be useful.
| The distinction between invertible elements (units) and
| irreducible elements, while complicated, also gave us a
| conceptual framework allowing us to prove lots of interesting
| and useful theorems.
| tshaddox wrote:
| It seems a little inconvenient to require acceptance that empty
| products equal 1, since that is also slightly subtle and
| deserving of its own explanation of mathematical terminology.
|
| Of course, I generally hear the fundamental theorem of
| arithmetic phrased as "every integer greater than one..." which
| is making its own little special case for the number 1.
| feoren wrote:
| >It seems a little inconvenient to require acceptance that
| empty products equal 1
|
| Only the contrary: it is extremely inconvenient to _not_
| allow the product of an empty sequence of numbers to equal 1.
| The sum of an empty sequence is 0. The Baz of an empty
| sequence of numbers, for any monoid Baz, is the identity
| element of that monoid. Any other convention is going to be
| very painful and full of its own exceptions.
|
| There are no exceptions to any rules here. 1 is not prime.
| Every positive integer can be expressed as the unique product
| of powers of primes. 1's expression is [], or 0000..., or
| [?].
| wesselbindt wrote:
| Any convention comes with the inconvenience of definition and
| explanation. So to call the convention that the empty product
| equals 1 based on that alone seems a bit unfair. The reason
| the mathematical community has adopted this convention is
| because it makes a lot of proofs and theorems a bit easier to
| state. So yes, you lose a bit of convenience in one spot, and
| gain a bit in a whole bunch of spots.
|
| And note that this convention is not at all required for the
| point I'm making regarding prime numbers. As you say
| yourself, restrict the theorem to integers greater than 1,
| and you can forget about empty products (and it is still
| easier to state if 1 is not prime (which it isn't)).
| dullcrisp wrote:
| This is like a "do arrays start at 0 or 1" question, except as
| they mention, algebraic number theory pretty much settles it.
| Whether 0 is a natural number though is still open for
| bikeshedding.
| fpoling wrote:
| I always thought that 0-based indexes were superior until few
| years ago I needed to deal with Fortran code and I realized
| that 1-based arrays allowed to use 0 as a non-existing index or
| sentinel, not size_t(-1) hack as found in C/C++. Like the
| article explains, depending on the domain one or the other
| convention can be advantageous.
|
| And then C/C++ compilers are subtly inconsistent. If 0 is valid
| index, then null should correspond to uintptr_t(-1), not 0
| address. That lead to non-trivial complication in OS
| implementations to make sure that the address 0 is not mapped
| as from hardware point of view 0 is absolutely normal address.
| IshKebab wrote:
| No, this article makes the case _for_ 0-based indexing. Let
| 's ignore the reality that computer fundamentally use 0-based
| indexes... The article says 1 is not prime because maths gets
| more awkward if it is.
|
| In the same way we index from 0 because indexing gets way
| more awkward if we index from 1.
|
| In-band sentinels are both quite rare, and also equally
| convenient with -1 or 0. In fact I would say -1 is a bit more
| elegant because sometimes you need multiple sentinel values
| and then you can easily use -2 (what are you going to use 0
| and 1 and then index from 2?).
|
| The more common operations are things like indexing into
| flattened multidimensional arrays, or dealing with intervals,
| which are both way more elegant with 0-based indexing.
|
| 0 is a valid index into an array. It's even a valid index
| into global memory in some environments. Not mapping memory
| to address 0 is completely trivial. I'm not sure what non-
| trivial complications you're thinking of.
| cogman10 wrote:
| I'm sure it depends on the definition of prime. I've always been
| partial to "Any integer with exactly 2 divisors". Short, simple,
| and it excludes 1 and negative numbers.
| JadeNB wrote:
| > I'm sure it depends on the definition of prime. I've always
| been partial to "Any integer with exactly 2 divisors". Short,
| simple, and it excludes 1 and negative numbers.
|
| Depending on your definition of divisor, it excludes everything
| _except_ 1 and -1, whose two integer divisors are 1 and -1. But
| then, if you specify that "divisor" means "positive integer
| divisor", it no longer automatically excludes the negative
| numbers, since the two positive integer divisors of -2 are 1
| and 2. (Incidentally, plenty of algebraists, myself included,
| are perfectly comfortable with including -2 as a prime.)
| mathemadigal wrote:
| I think we'll need to wait for an answer to if there is a prime
| number generating function.
|
| At that time we can determine if 1 is prime.
|
| If it's found that Eratosthenes' sieve is the only prime
| generating function then we have our answer.
| dullcrisp wrote:
| We'd have our answer in what way?
| mathemadigal wrote:
| I apologize for the ambiguity, it's apparent when you've read
| the article as it addresses this specific point.
|
| Namely, if the sieve is the only generating function for all
| of the primes then 1 would need to be omitted as prime as
| removing its factors would remove every number, thus failing
| to generate the list of primes.
| fhars wrote:
| If you treat one as prime number when running the sieve
| algorithm, one is the only prime number that remains after
| you have removes all its multiples from the list of candidate
| numbers.
| SkySkimmer wrote:
| Since 2 is prime 1, wouldn't it be more symmetric if 1 was prime
| 2?
| bluGill wrote:
| What makes you think two is prime? Not everyone would agree
| with that statement as the artical points out.
| spiderice wrote:
| The article states that historically Nicomachus of Gerasa
| didn't consider 2 a prime, in like 100 AD.
|
| Nowadays 2 is considered prime. Seems silly to question why
| someone is claiming 2 is prime if that is how it is defined
| in modern day.
|
| > What makes you think two is prime
|
| The current mathematical definition of a prime number
| dullcrisp wrote:
| Technically yes
| munchler wrote:
| Other good nerd-sniping math questions:
|
| 0^0 = 1? Yes, it's simpler that way.
|
| 0! = 1? Yes, it's simpler that way.
|
| 0/0 = [?]? No, it's undefined.
|
| 0.9999... = 1? Yes, it's just two ways of expressing the same
| number.
|
| 1+2+3+... = -1/12? No, but if it did have a finite value, that's
| what it would be.
| rvba wrote:
| If we try to define division by zero, shouldnt 0/0 be 1?
|
| Or even more abstract "every element on y". Which I think could
| sort of work
| meroes wrote:
| 0^0 got Gemini 2.5 pro the other day for me. It claimed all
| indeterminate forms (in the context of limits) are also
| undefined as a response to a prompt dividing by zero. 0^0 is
| the most obvious exception, it's typically defined as =1 as you
| said.
| robinhouston wrote:
| Another very interesting article on the primality of 1 is Evelyn
| Lamb's _Why isn't 1 a prime number?_
| (https://www.scientificamerican.com/blog/roots-of-unity/why-i...)
|
| A slightly facetious answer might be that this is the wrong
| question to ask, and the right question is: when did 1 stop being
| a prime number? To which the answer is: some time between 1933
| (when the 6th edition of Hardy's _A course in pure mathematics_
| was published) and 1938 (when the 7th edition was published).
| ks2048 wrote:
| Can we declare 2 composite? Kind of annoying to have an even
| number in there.
| dullcrisp wrote:
| It's composite in the Gaussian integers, maybe that helps.
| lern_too_spel wrote:
| Only if we can declare 3 composite because it's annoying to
| have a number divisible by 3 in the primes, and so on for the
| rest of them.
| vikingerik wrote:
| 2 being the only even prime isn't really anything fundamentally
| weird. Every prime is the only divisible-by-that-number prime.
| 2 has nothing unique about that.
|
| We only notice the case for 2 because our human languages
| happen to define divisible-by-2 as a word and concept. If our
| languages called divisible-by-3 "treven" or something like
| that, we'd think it weird that 3 was the only treven prime.
| jconder wrote:
| Odd to see an article about prime numbers with no mention of
| ideals. If (1) was a prime ideal then it would be the only non-
| maximal prime ideal. And it would be the only closed point in
| Spec(Z)...
| feoren wrote:
| All models are wrong, but some models are useful. It's not useful
| to consider 1 prime, so we don't. You're free to invent a new
| model of math where 1 is prime and see where it takes you; nobody
| will be offended. This happens all the time: "but what if we
| _could_ take the square root of a negative number? What then? ",
| etc. 99% of the time, this leads to a theory that is provably
| inconsistent and therefore useless. Out of the remaining 1%,
| about 99% of the time it leads to a mathematics that is simply
| less useful than what we have now. So it goes with making 1
| prime. Out of the remaining cases, about 99% of those turn out to
| be identical to an already existing mathematical theory, which is
| interesting (and possibly publishable), but not hugely useful.
| But about 1% of 1% of 1% of the time, these exercises result in
| actual new math that can tell us new things about reality and
| solve problems we couldn't solve before.
|
| This is not one of those times.
| rtkwe wrote:
| I've always wondered what actually breaks if 1 is prime or
| conversely what defining 1 as not prime gives us. Got just far
| enough into my math degree before switching to CompSci to stay
| of of universities the rest of my life to want to know.
| samf wrote:
| Some examples are in these comments, e.g. the Fundamental
| Theorem of Arithmetic. The Sieve of Eratosthenes is an
| amusing outcome, where 1 is the only prime if you take it
| literally.
|
| But also mentioned elsewhere in the thread: if we declared 1
| to be a prime, then many (I daresay "most") of our theorems
| would have to change "prime number" to "prime number greater
| than one".
| feoren wrote:
| The biggest problem is that you lose unique prime
| factorization. With prime factorization, I get a unique
| representation of every positive integer. Let's consider a
| way to write positive integers in "base prime", similar to
| base 10 or base 2. I'll start counting from 1 and write
| numbers as a tuple of prime factors. Similar to base 10,
| "base prime" has an infinite set of 0s that we're leaving out
| for brevity (e.g. 19 = 0000019), although it's on the right
| side instead of the left. 1 = () = (0, 0,
| 0, 0, 0, ...) 2 = (1) = (1, 0, 0, 0, 0, ...)
| 3 = (0, 1) 4 = (2) 5 = (0, 0, 1) 6 =
| (1, 1) 7 = (0, 0, 0, 1) 8 = (3) 9 =
| (0, 2) 10 = (1, 0, 1)
|
| The _i_ th position in every tuple is the power of the _i_ th
| prime in the factorization of that number. So 10 = (1, 0, 1)
| = 2^1 * 3^0 * 5^1. 84 would be (2, 1, 0, 1) = 2^2 * 3^1 * 5^0
| * 7^1. If we have unique factorization, there is exactly one
| way to write every positive integer like this, and there are
| many insights we can gain from this factorization. If 1 is
| prime, then we can write 6 = 1^257 * 2^1 * 3^1, or any other
| power of 1 we like. We just gain nothing from it.
|
| There are often many equivalent ways to define any
| mathematical object, and I'm sure there are plenty of ways to
| define a prime number other than "its only factors are itself
| and 1". These other definitions are likely to obviously
| exclude 1. One obvious one is the set of basis coordinates in
| this "unique factorization" space that I just laid out here.
| And we're never really excluding or making a special case for
| 1, because 1's factorization is simply the absence of any
| powers -- empty set, all 0s, whatever you want to call it.
|
| Keep in mind that "unique factorization" turns out to be very
| interesting in all sorts of other mathematical objects:
| rings, polynomials, symmetries, vector spaces, etc. They
| often have their own notion of "prime" or "primitive" objects
| and the correspondence with integer-primes is much cleaner if
| we don't consider 1 prime.
| samf wrote:
| This is the best answer.
|
| We could declare 4 to be a prime number, and keep the rest of
| the definition the same. Instead of just saying "no", you could
| ask, "okay, what would that do for us?" If there isn't a good
| answer, then what's the point? And usually, you're not in the
| 1% of 1% of 1%.
| alganet wrote:
| "Only divisible by itself and 1" is a darn elegant definition.
|
| 1, 2 and 3 are kind of special to me. In prime distribution
| studies, I discovered that they are special. It gets easier for
| some things if you consider primes only higher or equal to 5.
| Explaining distribution gets easier, some proofs become more
| obvious if you do that (tiny example: draw a ulam-like spiral
| around the numbers of an analog clock. 2 and 3 will become
| outliers and a distribution will reveal itself along the 1, 5, 7
| and 11 diagonals).
|
| Anyways, "only divisible by itself and 1" is a darn elegant
| definition.
| mikepurvis wrote:
| The 1 exception matters as well for prime mutuality, like X and
| Y share no common factors _other than 1 of course, sigh_.
| alganet wrote:
| I see 1 as mostly an anchor. However, my thing is not about
| working out axioms and formal mathematics. I do some
| visualizations that can help demonstrate aspects of prime
| distribution.
|
| I am fascinated by geometric proofs though. The clock thing
| is just a riff on Ulam's work. I believe there is more to it
| if one sees it as a geometric object and not just a
| visualization drawing. I could be wrong though.
| teytra wrote:
| When I was younger I had a period I often was thinking about
| prime numbers (before I got old and started thinking about the
| Roman Empire).
|
| I noticed the same as you, and IIRC the (some?) ancient greeks
| actually had an idea about 1 as not a number, but the unit that
| numbers were made of. So in a different class.
|
| 2 and 3 are also different, or rather all other primes from 5
| and up are neighbours to a multiple of 6, (though not all such
| neighbours are primes of course).
|
| In base-6 all those primes end in 5 or 1. What is the
| significance? I don't know. I remember that I started thinking
| that 2*3=6, maybe the sequence of primes is a result of the
| intertwining of numbersystems in multiple dimensions or
| whatever? Then I started thinking about the late republic
| instead. ;)
| alganet wrote:
| If you work not only the primes, but also the modulus
| function value of each non-prime, things get even more
| interesting than thinking of base changes! To me, it reveals
| much more.
| alganet wrote:
| Also, rearrangements.
|
| In two dimensions is easier.
|
| I cannot rearrange one pebble.
|
| I can rearrange two or three pebbles equidistant from each
| other in just one distinct way (inverting the position of a
| neighbouring pebble).
|
| And so on...
|
| There are many ways to think of natural numbers without
| actual numbers.
| scythe wrote:
| 1 is not a prime number because it would ruin the Euler product
| formula for the Riemann zeta function.
| EnPissant wrote:
| In programmer terms, imagine you had to define the product
| function in Python. The most natural way to write it is:
| >>> def product(ints): ... result = 1 ... for
| int in ints: ... result *= int ... return
| result
|
| In which case there is no need to make 1 a prime as you already
| have: >>> product([]) 1
| 2OEH8eoCRo0 wrote:
| 1 x 1 = 1
|
| 1 x 1 x 1 = 1
|
| ...
|
| Not prime!
| pwdisswordfishz wrote:
| > One way in which 1 "quacks" like a prime is the way it accords
| with Euclid's Lemma, the principle that asserts that if p is a
| prime, then whenever the product of two integers is divisible by
| p, one of the two numbers or both must be divisible by p.
|
| This is debunked by
| https://ncatlab.org/nlab/show/too+simple+to+be+simple#relati...
| pabenson wrote:
| Since 1 is the multiplicative identity (x * 1 = x for any x in
| the set) and any definition of "prime" needs to use
| multiplication then one way or another 1 is going to be special
| when talking about primes whether it is included in the set of
| prime numbers or not. You can't avoid 1 being "special"
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