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Why isn't 1 a prime number?
- earthicus 8y agoVery nice article! Also of interest, 1 not being prime can be viewed as a particular example of a more general phenomenon that occurs throughout mathematics, known as the 'Too simple to be simple' principle: https://ncatlab.org/nlab/show/too+simple+to+be+simple https://ncatlab.org/nlab/show/too+simple+to+be+simple
- bordercases 8y agoCould you have discovered the example through use of the principle? Or likewise, does the principle let you extend to other conclusions?
- earthicus 8y agoDo you mean the example of 1 not being a prime, or the example in the article of Z[root -5] not having unique factorization? You could certainly discover the former, but not the latter as far as I can see. The idea is that the principle should guide you on what the 'right' set of definitions are, so that your theory detects interesting theorems and isn't riddled with edge cases. You might make an analogy to programming, finding good primitive data types and interfaces so that you can implement elegant and efficient algorithms and well structured modules. Perhaps the key insight of the 'too simple to be simple' principle is that we want the definition to impose both existence & uniqueness of something. In this case proper divisors (divisors strictly less than the number itself e.g. the proper divisors of 6 are 1,2, & 3). ------------------------------- Let me give a concrete example of how it might be used. - Uniqueness is captured algebraically here by the notion of a 'subsingleton' set: a subset where any pair of elements are equal. This can happen if (1) the subset has only a single element, or (2) the subset has NO elements (in which case the uniqueness requirement is vacuously true). - Existence is captured by the notion of a 'singleton' set: a subset with exactly 1 element: any two elements belonging to the set are unique, AND such an element exists. First lets apply this to the definition of the regular primes: the set of proper divisors of 1 is the empty set. The set of proper divisors of any other prime is the singleton set {1}. Thus the naive definition says 'proper divisors of primes form a subsingleton': it asserts they are unique, but not that they exist. The more sophisticated definition of the primes (which excludes 1) asserts the proper divisors of primes form a singleton: uniqueness and existence. Now lets try to apply it (somewhat informally) to the fictional example of even numbers and 'even primes'. Examples: 2 is an even prime, 4 = 2*2 not an even prime 6 is an even prime, 8 = 4*2 is not, 10 is even prime, 30 is an even prime is as well, and so on. Here I have uniqueness of proper divisors (they form a subsingleton), but only because existence of proper divisors fails for every even prime (not just 2)! Now what happens if we try and factor 60? 60 = 2*30 60 = 6*10 It not great surprise that uniqueness of prime factorization fails as well (one of several problems with this informal example). The principle didn't help me find this example, it suggests that if I use the above notion of an 'even prime', i'm not going to get a good set of theorems.
- gabrielblack 8y agoBad example, if 1 isn't prime, 2 x 1 isn't a product of prime, so not all the number can be written as product of primes. My point is that you can't write an introductory article, supposed to be understandable to the masses, using a form able to confuse further the readers. A student, reading this, I don't think will understand this explanation that seems contradictory. "My mathematical training taught me that the good reason for 1 not being considered prime is the fundamental theorem of arithmetic, which states that every number can be written as a product of primes in exactly one way. If 1 were prime, we would lose that uniqueness. We could write 2 as 1×2, or 1×1×2, or 1594827×2. Excluding 1 from the primes smooths that out."
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- kdmccormick 8y agoAll positive integers can be written as a product of primes, where a product is a sequence of numbers that are multiplied together. In the case of a composite number like 45, the primes are (3,3,5). When the sequence has one element, then that element is the product. So, 2 is the product of (2). When the sequence is empty, the product is 1.
- theoh 8y agoThis is explained more formally here: https://en.wikipedia.org/wiki/Empty_product https://en.wikipedia.org/wiki/Empty_product
- skoskie 8y agoAs a total layman, I think this is my favorite definition so far.
- mintplant 8y agoFurthermore, the ordered sequence of primes that multiply to any given positive integer is unique. If 1 were considered prime, that wouldn't hold, as you could keep adding as many 1s as you like to the start of the sequence: (1,3,3,5), (1,1,3,3,5), etc. Thus 1 encodes no useful information here.
- bensalem 8y agoJust the definition. a number with no factors beside 1 and itself, excluding zero 1 does not meet that definition, since 1 is already counted, the other factor needs to be different.
- mcphage 8y agoSure, but the next question—which the linked article is about—is, why is that the definition?
- bensalem 8y agopedant. because it is a unit. but one thing I think is interesting is you can further categorize primes. FTA is unique factorization. but those factors can be re-arranged (most FTA text says ignoring order). considering order is interesting. then you can class primes by how many ways they can be rearranged. in that case, including 1 in the factorization once is interesting, since it differentiates between squares n x n and cubes n x n x n which can not be rearranged anyway unless you include 1 (not as a 'prime' but as the unit in the factorization) 1 x n x n, n x 1 x n, n x n x 1 3 ways. And cubes, 4 ways. From that point of view, primes are simply the numbers that can only be rearranged 2 ways. 1 x p, and p x 1 This is a very interesting avenue.
- ShamelessC 8y agoThe article mentions some of the historical context around defining 1 as a unit and also suggests its usefulness as a number due to base 10 arithmetic. I also think it is interesting to call the article pedantic and then make a similarly pedantic argument just beneath that claim.
- rat9988 8y agoIt's against the guidelines to say it I think. But I'm pretty sure at this point he didn't read the article. He has just read the title and thought he had enough knowledge on the matter to answer it without reading the author's writing.
- HocusLocus 8y agoThe number 1 is a prank by Loki.
- frankbreetz 8y agoFunny true story about this: One time a man came up to me and handed me his phone, and asked me to call his mom. He was about to pass out and he asked me not call an ambulance (God bless America) , he appeared to have a concussion. By this time there was about 10-15 people around him and he could barely talk. I asked him what his phone code was and instead of just giving it to me he said "it's the first four prime numbers". Immediately, about five people shout "1,2,3,5". I am no longer holding the phone, because I handed to someone else to make sure it was okay. Sure enough, I was in a mathematical proofs class and we had just discussed this topic. So, I say "one is not a prime number". Of course, we get the phone unlocked in the second try with "2,3,5,7" and the guys mom is on the way. Everyone thought I was a genius a hero.
- rytill 8y agoThat’s great. Thank you for sharing. Seems a rather cryptic way to go about things with your health on the line.
- kzrdude 8y agoIt reveals how he thought about his code, and he didn't have the presence to "compute" what it would be in digits.
- frankbreetz 8y agoYeah, he was basically laying on the ground and barely got the phrase out.
- hackerman12345 8y agoReminds me of The Riddler...
- frankbreetz 8y agoHaha, This was the closest I've ever felt to being Batman
- myself248 8y ago
- blu42 8y ago"Whenever a product m×n is divisible by p, then m or n must be divisible by p." Actually it's the other way around: 'Whenever m or n are divisible by p, then their product m×n must be divisible by p.' The opposite is not true, i.e. the statement is not reversible and still true. For instance: 6x4 = 24; 24 is divisible by 8, i.e. 24 mod 8 = 0, yet 6 mod 8 = 6, and 4 mod 8 = 4. [ed] I've stepped into a cognition discontinuity, move along, nothing to see.
- jcranmer 8y agoIf p is a prime number, then the product m×n being divisible by p means that either m or n is divisible by p. In fact, this is how prime numbers are defined when generalized to generic integral domains.
- blu42 8y agoDoh, my reading balked at the embedded add. Yes, p being a prime actually inverts the statement successfully. My bad.
- camjw 8y ago8 is not prime
- pvg 8y ago"M₆₇ is not a prime" was also a famous single-slide silent lightning talk. https://en.wikipedia.org/wiki/Frank_Nelson_Cole https://en.wikipedia.org/wiki/Frank_Nelson_Cole
- crazygringo 8y agotl;dr: "...whether or not a number (especially unity) is a prime is a matter of definition, so a matter of choice, context and tradition, not a matter of proof." In other words, because semantics.
- fjsolwmv 8y agoThat shows the author doesn't understand the issue well, because every definition is semantics. That's what the word means. The question is why certain semantics are prefered.
- Hackbraten 8y agoAnd, even more important: is it Numberwang?
- HenryBemis 8y agoWikipedia [1] also mentions that: "Most early Greeks did not even consider 1 to be a number, so they could not consider its primality". Both articles (scientific american & wikipedia) refer to the same paper [2]. [1]: https://en.wikipedia.org/wiki/Prime_number https://en.wikipedia.org/wiki/Prime_number [2]:https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.html https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.h...
- empath75 8y agoI always raise an eyebrow at statements like ‘most early greeks’, because who knows how many of them actually read Euclid or had any exposure to math at all.
- pcmaffey 8y agoI've always thought of 1 as the definition of what a prime number is. Therefore, because it's generally bad practice to define a thing in terms of itself, 1 shouldn't be categorized as a prime number. Instead, it's like the Aristotilian Ideal of prime made linear.
- dmarlow 8y agoInteresting read. I'm an engineer and had considered 1 to be prime. I guess you can say that 1 is just an absolute unit.
- antaviana 8y agoI understood prime as the minimal set of numbers with which you can generate all natural numbers by multiplying them. There are infinite natural numbers and infinite prime numbers, but you need each and every prime number to generate any other natural number. So I'm puzzled to learn that 1 is not a prime number because I do not know how to generate 1 if we exclude 1 from the prime set.
- ummwhat 8y agoFor that matter, even with 1, how do you generate 0?
- roywiggins 8y agoZero being a member of the natural numbers or not depends on convention.
- traderjane 8y agoI would argue that Zero is generally considered to be a natural. The naturals are often constructed via an initial object, and you can call that initial object whatever you want (some people write Z or Zed), but it's metaphorically a zero.
- rocqua 8y agoThe pushback I have gotten om my tongue-in-cheek rants defending 0 as a natural number would suggest not everyone agrees. Some people really care about 1/n being possible for every natural number N.
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- bo1024 8y agoI was glad that the article got into the definitions of irreducible, prime, and unit in more general algebraic structures! This question is all about these three concepts. A unit is an operation you can apply multiple times and get back to the starting point, for example multiplication by 1 or -1. So these are units in the integers. Units are really important, but they don't get you anywhere on their own. We can think of non-units as building blocks. So then the question comes up which building blocks can be broken down into smaller blocks, like "times 6" can be divided into "times 2, then times 3". Using a unit doesn't count, because you're still equally far from the starting point (so 6 = (-6)(-1) is not really breaking down 6). Blocks that can't be broken down any farther are irreducible (like 7). But in common parlance, that's what we think of as the definition of a prime number. It doesn't include units like 1, because they're already broken down. More generally, being prime isn't about being broken down, it's about building up. Suppose I tell you that n is divisible by 2, and n can be broken down into two pieces, n = ab. Then you know that at least one of a or b is divisible by two. So two is prime because anything that is built from two, when broken down into pieces, has a piece that can be built from two. (Example: 6 divides 12, but you can break 12 up into (3)(4), so 6 isn't prime.) This definition of prime also doesn't include units like 1 because they're not building blocks in the first place. Everything's already divisible by units. Prime integers have both the prime property and the irreducible property. Either way, I think it makes sense that 1 isn't included -- being prime and irreducible are properties of building blocks, and 1 is a unit.
- red_trumpet 8y ago> A unit is an operation you can apply multiple times and get back to the starting point, for example multiplication by 1 or -1. Nope. That is an idempotent[1], or a root of unity, i.e. x^n = 1 for some n >= 0. A unit is something which has an inverse, though that inverse need not be a power of x itself. For example, 1/2 is inverse to 2 in the rational numbers. [1]: https://en.wikipedia.org/wiki/Idempotence https://en.wikipedia.org/wiki/Idempotence
- bo1024 8y agoOops, thanks! (been a while) Those are examples of units, but units are more generally "building blocks of 1" (if you apply a unit, there is some other unit that gets you back to 1). Thanks again for the correction.
- legohead 8y agoWhen attempting to prove intelligence to aliens by mathematical proof, would we include 1 as a prime number?
- mcguire 8y agoMathematicians: No! Engineers: Eh, maybe? The guy doing the engraving on the gold plaque on the side of the space probe: What is that, a potato? Aliens: They treat root vegetables as numbers. Alien anthropologist: They worship plants. Alien TV presenter: In showing us agriculture, they introduced us to mathematics! (Me: I think I need to cut down on the cold medicine.)
- fjsolwmv 8y agoIt doesn't matter. An additional element at the beginning of a sequence won't make anyone think you unintelligent, just a bit confused
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- jmmcd 8y agoAnyone who says 'zed' and indeed cheerio, pip pip, won't say "negative five". They'll say "minus five", of course.
- ColinWright 8y agoThat depends - this divide is not just "USA English Speaking People" versus the rest. Within the UK, Australia, New Zealand, and Canada, there is a split between people who say "minus 5" and people who say "negative 5". For some "minus five" is an operation, not a number, and for others "negative five" is meaningless, the correct expression would have to be "the negative of five". There are entire PhD theses written on the topic of how to avoid confusing young children when introducing them to numbers less than 0, and the words we should, and should not, use in that context.
- jmmcd 8y agoI was thinking more of UK (the only place where "pip pip" makes sense as a stereotype) versus the rest! Are you saying there are those in the UK who say "negative five"? That would surprise me.
- ColinWright 8y ago> Are you saying there are those in the UK who say "negative five"? Yes, there are.
- gerdesj 8y agoYes, and who on earth says "cheerio, pip pip"? "Toodle pip" or "cheerio" or "pip pip" are acceptable. Those PhD theses seem a little misguided (if well meaning) to me. Language is riddled with ambiguity - "panda eats shoots and leaves" for example. Despite that ambiguity we generally manage to get along. Teaching critical thinking in general might be a better idea than fixating on specific issues.
- ColinWright 8y ago
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- rdl 8y agoThere are also a bunch of cases where you specifically want "odd primes" (i.e. excluding 2).
- Someone 8y agoThe problem here is that “is not prime” doesn’t imply “is composite”. The number 1 is neither prime nor composite. Why? Because that is the choice that makes more math look nicer. If 1 were considered a prime, lots of theorems would have to say “let p be a prime >1”, rather than “let p be a prime”. Mathematicians have struggled with this for centuries before agreement was reached. See “The History of the Primality of One: A Selection of Sources” (https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.pdf https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.p...) The “math looks nicer” argument isn’t unique to this example. It also is, for example, the ‘reason’ that 0⁰ equals 1, or that there are as many integers as rationals.
- pvg 8y agoIn what way is the last one a matter of making it 'look nicer'? Either there is a bijection between these or there isn't - that seems like a concrete thing that's not a choice of terminology or aesthetics.
- Someone 8y agoBy making the choice that that applies to infinite sets, too, you go against strong intuition such as that, for example, there are twice as many integers as even numbers (that intuition is very strong; even math students will start out thinking it is true, not “I wouldn’t know”)
- nimih 8y ago> The “math looks nicer” argument [...] is, for example, the `reason' [...] there are as many integers as rationals. I'm curious what you mean here: The integers and rationals have quite different algebraic and geometric properties, and the fact that they have the same cardinality has always struck me as a deep result, rather than just a matter of definitions.
- pvg 8y agoJust occurred to me they were perhaps thinking of https://en.wikipedia.org/wiki/Continuum_hypothesis https://en.wikipedia.org/wiki/Continuum_hypothesis and mis-remembered/typed.
- hackerbrother 8y agoIt's just a definition!
- mikorym 8y agoAs the article says, one could consider this a matter of definition. The definition of choice for me (and some other category theorists?) is that the prime numbers are the second row in the devisibility lattice. (That is, n < m iff n | m.) Then, the number 1 is the single element of the first row and all the prime numbers are the atoms immediately above 1.
- waffleguy 8y agohonestly this is where math stops being logical and follows emotions. The logical reason one isn’t prime is because we don’t want to ruin theorems that were based on faulty logic so we continue with the faulty logic... yay?
- FartyMcFarter 8y agoShort answer: 1 could be counted as a prime number, and in fact it used to be counted as a prime number by some mathematicians [1]. These days, most people just choose not to do that, since this makes most proofs and definitions more convenient. Shorter answer: Convenience. [1] http://mathworld.wolfram.com/PrimeNumber.html http://mathworld.wolfram.com/PrimeNumber.html (3rd paragraph)
- arithma 8y agoI am not a mathematician. I reached solace regarding this itchy issue (not even joking) by considering the following: Primes are the minimal set of numbers to generate another set of numbers, using multiplication (with repetition, yada yada) For the positive numbers, [2, 3, 5, 7, ...] is all we need, considering that, as many others have said, 1 = Product([]). In that sense, to generate all integers, the integers you need are: [-1, 0, 2, 3, 5, 7..] Similarly, to generate all gaussian integers (complex numbers with integer components,) you can follow the following link: https://en.wikipedia.org/wiki/Gaussian_integer#Gaussian_primes https://en.wikipedia.org/wiki/Gaussian_integer#Gaussian_prim... however, am not sure why 0 is not considered a prime number in this context. Gaussian primes are symmetric over the real and imaginary axes, and to me it's quite curious why 0 is not considered prime there, but I guess uniqueness of the multiplication matters too (0 can be repeated multiple times, which could make some theorems about uniqueness of factorization a bit cumbersome as well.)
- rocqua 8y ago> I am not sure why 0 is not considered a prime number in this context. So, if you have a set with multiplication and addition, that is called a ring. A very nice class of rings is called fields. This is where division is possible. Another way to state that 'division is possible' is to say that multiplication is a group operator on the entire set except zero.
- contravariant 8y agoYou might be happy to know that 0 is considered a prime element in typical cases (barring those cases where there are 0 divisors, i.e. where two elements can be multiplied to create 0, obviously this is not an issue with regular numbers). Edit: for historical reasons prime elements are slightly different from the usual prime numbers, and 0 remains a bit of a special case even now.
- yakubin 8y ago> I am not sure why 0 is not considered a prime number in this context. The greatest common divisor of any two primes is one. To say that two numbers are coprime, is to say their greatest common divisor is 1. If we considered 0 a prime, it would be a bit weird (and messy) that two prime (0 and any other prime number) numbers aren't coprime.