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Is 1 Prime, and Does It Matter?
- pyfon 1y agoWhat does primarily look like with the addition operation instead of multiply? 1,2,4,8,...? Or indeed just 1 alone lol! (Yes 1 is there because zero is the additive identity)
- scythe 1y ago1 is not a prime number because it would ruin the Euler product formula for the Riemann zeta function.
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- JJMcJ 1y agoOne 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 1y agoYes, 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 1y agoI'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 1y agoCorrect, 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 1y agoDo you have a link to where I could learn more about this?
- gerdesj 1y agoYou might start here: https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach That's the GEB I mentioned above.
- pinkmuffinere 1y agoI 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 1y agoAs 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.
- seanhunter 1y agoExactly. This is similar to the case of how the zero function provides a trivial solution to almost every differential equation.
- JadeNB 1y ago> 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-examples-of-vacuous-void-entities https://mathoverflow.net/questions/45951/interesting-example....)
- tux3 1y agoThat'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 1y ago> 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 so may indifferently be denoted by *; and then charge ahead with the construction, using not Ø, Ø ∪ {Ø}, Ø ∪ {Ø} ∪ {Ø ∪ {Ø}}, 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.
- tikhonj 1y agoAnd 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 1y agoTo 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-odd-prime https://math.stackexchange.com/questions/1177104/what-is-an-...
- kordlessagain 1y agoThe 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 1y ago> 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 1y agoIt's actually the least odd prime
- chrismcb 1y agoIt 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..."
- arcastroe 1y ago"2 is the only even prime number. Therefore, it's the oddest of them all!"
- bluepnume 1y agoIt's hardly odd. "Even" just means "divisible by 2" "2 is the only prime that is divisible by 2" "3 is the only prime that is divisible by 3" "5 is the only prime that is divisible by 5" ... "N is the only prime that is divisible by N"
- mystified5016 1y agoSeems that if we must add all these conditions to make the definition of prime consistent, maybe we shouldn't consider it prime?
- wesselbindt 1y agoIf 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 1y agoNot just that one; practically every useful theorem about primes would have to be rewritten to "if p is a prime other than 1".
- SketchySeaBeast 1y agoIsn'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 1y ago1 is not greater than 1, and a product of one prime is still a product of primes
- SketchySeaBeast 1y agoYeah, I didn't understand you can have a product of a single number.
- jdoliner 1y agoMathematicians 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 1y ago
- dullcrisp 1y agoThis 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 1y agoI 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 1y agoNo, 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 1y agoI'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 1y ago> 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 1y agoI 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 1y agoWe'd have our answer in what way?
- mathemadigal 1y agoI 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 1y agoIf 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.
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- SkySkimmer 1y agoSince 2 is prime 1, wouldn't it be more symmetric if 1 was prime 2?
- bluGill 1y agoWhat makes you think two is prime? Not everyone would agree with that statement as the artical points out.
- spiderice 1y agoThe 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 1y agoTechnically yes
- munchler 1y agoOther 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 1y agoIf 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
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- croes 1y agoBut that would mean (0/0) * 2 = 2 but (0/0) * (2/1) = (0 * 2) / (0 * 1) = 0/0 = 1
- meroes 1y ago0^0 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.
- margalabargala 1y ago> 1+2+3+… = -1/12? No, but if it did have a finite value, that’s what it would be. The other ones, sure, but I'm not following this one.
- munchler 1y agohttps://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B...
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- robinhouston 1y agoAnother 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-isnt-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 1y agoCan we declare 2 composite? Kind of annoying to have an even number in there.
- dullcrisp 1y agoIt’s composite in the Gaussian integers, maybe that helps.
- lern_too_spel 1y agoOnly 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 1y ago2 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.
- dullcrisp 1y agoIt’s a little weird. Numbers being their own additive inverse in characteristic-2 makes for some special cases. (But I guess if we did algebra with ternary operators, 3 might be weird too.)
- jconder 1y agoOdd 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 1y agoAll 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 1y agoI'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 1y agoSome 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".
- alganet 1y ago"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 1y agoThe 1 exception matters as well for prime mutuality, like X and Y share no common factors other than 1 of course, sigh.
- alganet 1y agoI 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 1y agoWhen 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. ;)
- EnPissant 1y agoIn 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 1y ago1 x 1 = 1 1 x 1 x 1 = 1 ... Not prime!
- chris37879 1y agoDepends on your definition of prime, by your reasoning, I could say 7 * 1 * 1 = 7, so it's not prime. Better to say a prime is any number with a set of divisors of length 2 including 1 and itself. If you want to exclude 1.
- pwdisswordfishz 1y ago> 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#relationship_to_biased_definitions https://ncatlab.org/nlab/show/too+simple+to+be+simple#relati...
- pabenson 1y agoSince 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"
- scoofy 1y agoJust a note from your friendly philosophy degree holder: Axioms are arbitrary. Use the axioms that are the most useful.
- Spivak 1y agoDefinitions are neither true nor false. They're either useful or not useful. The question of whether or not the integer 1 is a prime doesn't make sense. The question is is it useful to define it as such and the answer is a resounding no.
- gblargg 1y agoAgreed. Definitions are made to differentiate things in a way useful for some goal. The question "Is X an M?" without a context or goal basically picks up whatever vague goals or purposes a person has lingering below the surface of consciousness, differing from what other participants have below theirs, leading to different answers, with no way to select the best one. In the case of what is considered prime, it's a matter of what definition simplifies the things that use it. It could be that two concepts are better, one including 1 and the other not including it. Since it's just a language shorthand, it makes no fundamental difference other than efficiency and clarity in communication about math.
- NoahZuniga 1y agoWhile axioms are in some sense arbitrary, it is helpful if they are consistent (informally: you can't prove something that "is false"; formally: you can't prove p and not p). Also other people like it if your axioms feel obvious.
- scoofy 1y agoMy point is that axioms "feeling obvious" is exactly a signal that they will be useful. The point of deductive reasoning based on axioms is that it is a shortcut to fill in problems of induction, which is what happens when we use pure empiricism. If you really want to go down the road of solipsism, read Karl Popper.
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- sepidy 1y agoI think 1 is so different from other numbers, it seems that in the past, some people did consider 1 to be a prime number. However, by the early 1900s, mathematicians agreed to exclude 1 from the list of primes to keep mathematical rules clear and consistent.
- 77pt77 1y agoIf it is unique factorization in terms of prime numbers goes out of the window and that is the main reason it usually isn;t considered.
- amavect 1y agoSome other definition fun: Should we define 0 both positive and negative, or neither positive and negative? Does monotonically increasing mean x<y -> f(y)<f(x) or x≤y -> f(x)≤f(y)? Should we deny the law of excluded middle and use constructive math? Does infinity exist? If infinity exists, is it actual (as an object) or potential (as a function)? Is the axiom of choice true? Or, is the axiom of determinacy true? Should we use a space-time manifold, or separate space and time dimensions? Do future objects exist, and do past objects exist? Do statements about the future have a definite truth value? Does Searle's Chinese Room think? Which Ship of Theseus is the original: the slowly replaced ship, or the ship rebuilt from the original parts? I find that so many philosophy debates actually argue over definitions rather than practical matters, because definitions do matter. Well, add your own fun definition questions!
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- LegionMammal978 1y agoWhat's worse, French typically uses positif to mean "greater than or equal to 0", so some people will act confused if you use English 'positive' instead of 'strictly positive' to mean "greater than 0".
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- AStonesThrow 1y agoI've been fascinated by numbers lately, and one of my go-to tools is a simple mobile app that calculates all the divisors of a given number. So I can determine prime numbers, and readily factor the non-primes. And it's been eye-opening. Now I'm no crackpot numerologist, adding up the numerical values of Bill Gates' name, or telling you who shot JFK. But I can tell you that the main launch pad 39A at Cape Kennedy was not numbered by accident -- look it up in the Book of Psalms. And it's interesting how the city buses around here are numbered. For example, the 68xx series; I look up Psalm 68 and I can definitely imagine the bus singing that as it lumbers down the road -- can't you? Back to primes -- if we consider the top numbers authorities of our times, such as the US Post Office, city planners, and the telephone company (circa 1970s). I ran a chunk of ZIP codes from Southern California and discovered that some are the factors of two quite large prime numbers. Others yield interesting factors. Once again I pull out my Book of Psalms. There are plenty of other "hermeneutics" to interpret assigned numbers, especially street addresses. And as for phone numbers, I've gone back to figuring out "what do they spell" on a standard TouchTone keypad, because sometimes it's quite informative. It's no accident, for example, that the hospital where I was born is located at 4077 5th Avenue. And that number assigned by city planners, many decades before M*A*S*H was written or went on TV. Significant nonetheless. I also figured out a few prime numbers related to my own life, and others that are recurring tropes, just cropping up at interesting times. What's your social security number? Have you sort of broken it down and pondered if those numbers turned up again and again in your life? Every time I see a number now, I'm compulsively factoring it out in my head. Is it prime? It feels prime. I'll check it in the app later; try some mental math for now. I'm also counting things more often now. How many spokes in a wheel? How many petals in a flower, especially a flower depicted in art. How many brick courses in that interesting wall they built? Plug any interesting numbers back into the divisors app. Finding the primes, find the factors, just ponder numeric coincidences. It's fun. So many signs and signals, hidden in plain sight before us. Buses singing Psalm 68 as they take on passengers. Launch pads singing Psalm 39 as Europa Clipper slips the surly bonds of Earth. What's on your telephone dial?
- hacksaurusRex 1y agoI think the main issue with "1" being prime is that without "1" each positive integer can be uniquely decomposed into a product of prime numbers. It is probably the most important fact about primes, and in this context "1" does look like an imposter.
- ryao 1y ago1 is not a prime number because Disquisitiones Arithmeticae did not regard it to be a prime number, and that book is the basis of number theory.