I've always rationalised it even further by thinking that 1 is a sort of "superprime", which is sort of true because it is the multiplicative identity. We have to separate it from the normal primes because it screws with a lot of theorems involving prime numbers.
I'm glad you're not strictly talking about math problems but also about the history and philosophy behind how we define things. And a video on deductive reasoning? Everything I ever wished for from a math channel.
In the Contact movie/book it makes some sense to include '1' since the aliens were replying to early radio and TV emissions from 1936 when possibly the 1 was listed as a prime by some sources as you just shown. Also, it makes sense to start a sequence with the fundamental unity of measurement in the same way that the messages on the Voyager included distances based on a multiple (in base 2) of the fundamental frequency of the hydrogen atom.
@@MagicGonads But this needs a historian of mathematics. If you read old mathematical papers, by the best mathematicians in history, you find that there was little agreement over whether to count 1 as a prone or not. There is agreement *now*, but as recently as my schooldays in the 60s, our textbooks still included 1 in the primes. And they weren't wrong, they were just on the verge of being rendered out of date.
@@digitig the historical context is largely irrelevant because the term "prime" is also irrelevant, it's the concept underlying that term that we need in order to address the question in the contemporary language of mathematics (the context in which the question is asked). Just because historically "0" wasn't considered a number, does not mean the answer to the question "is 0 a number?" is "well we need to do a historical overview...", the answer is "it's an element of Z which are an archetypal example of a number set", the historical context may give you insight into why we currently consider 0 a number, but it won't decide the answer to the question. But, there are actually matters of convention in dispute, like "does N (the natural numbers) contain 0?", "does the cup (subspace relation) include proper subspaces?", it does require extra clarification if you use the symbol in a formula where it might matter (but usually there's an expansion of the formula included in the relevant working/proof anyway that clarifies that instantly) Some actually meaningful disputes are "most natural foundation / topos / calculus / axioms / construction?" which are things like 'intuitionistic' vs 'structuralist', that inform how you think about what underlies the mathematics you are doing (e.g. decides what standard of proof you stand by) Contemporary mathematics is all about taking our disagreements as possible pathways to new mathematics, constraints and the lack of constraints both lead to new ideas. We would not be saying "1 is not prime because 1 is not a number", there would be a reason based on the properties of the underlying theory we have chosen to work with to decide this.
I have come across it when determine size of various groups, such as U(20) would be 20 - 8 = 12 elements U(n) groups are groups when you pull out the numbers that are not relatively prime to n. And this group is a group under multiplication.
I feel like the totient function excluding 1 isn’t because 1 doesn’t work as a prime, but because it isn’t being counted as a factor. If you count all the numbers less than 20 that share a factor and you consider 1 to be a factor, then there are none because 1 is a factor of everything.
There's another definition of the totient function that is a little more structural and clearly doesn't depend on this choice. The totient of n counts all the numbers less than n which are units mod n. That is, it counts the numbers m < n for which there exists an associated integer d such that dm = 1 mod n, meaning that dm - 1 is divisible by n. This definition is clearly independent of whether you decide 1 is prime or not, and then including 1 as a prime totally messes up the count as shown in the video.
@@angelmendez-rivera351 I’ve seen you typing on a lot of these comments and I just have this to say: If I say all green animals should be considered plants because they have the same color, the current definition of “plant” is irrelevant because we are discussing the definition itself. Definitions are non-universal, made up by humans and always changing. Therefore you can discuss hypothetical scenarios were the current definition does not exist.
@@Alex-02 *If I say all green animals should be considered plants because they have the same color, the current definition of “plant” is irrelevant because we are discussing the definition itself.* If you say all green animals should be considered plants because they have the same color, then this does not address the current definition of "plant" at all, and as such, it does not replace said definition. Also, you chose a poor example for an analogy, because not all plants are green. In any case, if you want to define categories of objects by their color, then all you have defined is the color itself. This has nothing to do with biology. You can even define a subclass of the category of living beings based on color. Again, though, this has nothing to do with biology: this is simply about the color of the living beings. You are still including green fungi, green protozoa, green bacteria, etc., all in this category. There is no biological property that green animals share with plants, such that no other entities share that property, living or non-living. Therefore, it is ontologically inadequate to insist that they are in the same category. *Definitions are non-universal, made up by humans and always changing.* Well, this is just a false belief. Definitions are not mere labels you come up with arbitrarily on the basis of preference for labeling objects of your preference. Definitions have to be consistent with reality. To apply definitions to a set of properties, you first have to ensure that these properties actually are well-defined, and that there exists at least one object which has those properties. Then, when you apply the label to this set of properties, it follows that the label applies to _all_ objects satisfying these properties. If you want to single out certain such objects as being qualified, and excluding the rest, then you need to specify properties which are satisfied exactly only by those objects you are trying to single out. Also, for this to be justified, you cannot do this on an ad hoc whim. Otherwise, that renders all definitions as completely redundant and useless. What makes definitions useful is precisely their non ad hoc nature. Finally, there is the problem of decidability. A definition has to be decidable, because if there exists no algorithm by which anyone would ever be able to tell if a class of objects satisfies the given definition or not, then it is completely pointless, and as such, the 'definition' does not actually define anything. How precise you want to get with all of this, it all depends on what academic discipline of study you are a part of, or whether the definition is just intended for colloquial nonsense. In mathematics, though, the highest level of achievable precision with these definitions is required. On the note of universality, mathematics actually are universal. Mathematics are independent of culture, nation, religious belief, etc. Yes, the types of notation you use to talk about mathematical concepts vary from language to language and culture to culture. The mathematical concepts themselves, though, are universal. A quasigroup is defined in Norway in exactly the same way it is defined in South Africa. A topological space is defined in Malaysia in exactly the same as it is defined in the United States of America. The works that you see published by mathematicians from New Zealand are not in disagreement regarding the concepts and their properties with the works published by mathematicians in China. You seem to insinuate that mathematical definitions are on the same caliber as definitions of words you find in the Urban Dictionary. Yes, those words do vary in definition from location to location, and even just from person to person, and really, those words are not defined in any particular way at all, they are used arbitrarily, because they are not used to discuss concepts, they are just used to communicate basic bits of information about a particular ill-defined thing. This is not how mathematical definitions work at all, though. *Therefore you can discuss hypothetical scenarios were the current definition does not exist.* I can guarantee you that there is no hypothetical scenario in which the current definition would not exist and in which people have the knowledge of ring theory that we do today.
Thank you for resolving a 50+ year annoyance! I learned some maths before formal schooling from some of my dad's 1930s textbooks: 1 was listed as a prime. Then I went to secondary school and got (literally!) knuckle-rapped for answering 1 was prime, because the "new maths" had decided 2 was the first prime. The discrepancy has always irritated me ever since. Your explanation from Gauss finally makes me happy! Nice video: thank you.
Oh man, "until next prime" is a great closing stinger. If you used it just as a one-off for this episode, I urge you to consider making it your standard closing line.
I think what I find so appealing about your videos is that they tend to take ideas that a lot of mathsy people are probably familiar with (the representation of numbers as sets, the idea that one is not a prime, etc.) and shine a light on why we got there, rather than just the nuts and bolts of how those ideas work. I hope this isn't accidentally an insult, but I feel like the most enticing part of your videos has been that they straddle the border between mathematics and philosophy, albeit clearly on team mathematics. I think there has been some exploration of this space, but usually not with the level of sincerity and depth that you bring. Once again, an exceptional video!
the still ongoing disagreement over the inclusion of 0 in the "natural numbers" is extremely annoying, i hope we solve it soon enough that we don't have to constantly use the unambiguous "positive integers" and "nonnegative integers" all the time
What convinced me that 1 isn't a prime was the logarithmic integral curve matching pi(n), that was like Nature telling us that 1 isn't a prime. My favourite definition is "a number that has 2 unique factors" (no further clarification needed, provided that 1 and itself count as factors). 1 only has 1 unique factor, which means it has even less factors than a prime. Also the least number of factors of any integer (0 and infinity have infinity factors).
One way to justify the non-primality of 1 is to categorize natural numbers by their order in terms of divisibility. We say a | b ("a divides b") iff there is a natural number n so that an = b. Then we say a < b iff a | b and not b | a. Then you can verify that < is a strict partial order over N, and moreover, that 1 is the least element and 0 is the greatest element. That is to say, for any natural number n, 1 | n and n | 0. For any prime p, we have 1 < p, but there is no other number n for which n < p. So if you draw a graph of
It's a bit like defining 0! = 1 - We just do it because the alternative is to have to keep track of the special case of 0 (or 1 for primes) at every stage of a proof.
If you're writing a proof you get to decide whether 1 is a prime number for the purpose of that proof. So many things were defined for just one proof and then the definition stuck
Another Roof, you have been selecting some awesome math topics! To create some connective tissue, I noticed the Euclidian definition of the unit and then the number as a collection of units sounds a lot like the foundation for the "empty set" definition of numbers you outlined in a previous video. I didn't realize that method of number definition had such old historical precedent... Anyway, looking forward to future videos. Keep up the good work!
Until I started looking into it, I had no idea how long the "1 is its own thing and not a number" idea persisted! My next video will be something very different.
Great video. I really liked how you used this topic to talk about definitions and properties more broadly. I like how even if you’re talking about a ‘simpler’ topic you try to find the conclusion that helps a student to learn about math more broadly
AAAAA i'm so glad you mentioned the distinction of primes/units in general rings like the integers! That's what I was watching for the whole time lol, definitely wish you mentioned a bit more about it but fantastic video as always!
@@AnotherRoof I just binged your channel (+subscribed!) and from the Q&A you 100% seem like my kind of person, lol, so i have no doubt whenever you get around to it it'll be absolutely brilliant!
@@lexinwonderland5741 I agree with you that more should have been mentioned on it, since the distinction between irreducible elements and units in a ring is ultimately at the root of why 1 cannot be considered a prime number. Look, do not get me wrong. I think that understanding how mathematical concepts from antiquity involved into mathematical concepts today as our understanding of mathematics improved and became more refined is very fascinating, and certainly an important kind of knowledge to have in general. However, as far as answering the question "is 1 a prime number?," the history is not enlightening at all: it ultimately does not answer the question. Yes, I know that mathematicians in the 1700s thought of 1 as a prime number, this is all well and fine, but that tells us nothing as to whether 1 actually is or should be considered a prime number or not. These questions are questions regarding the relationships between various mathematical concepts at a foundational level, not questions about names and conventions that mathematicians vote on. If you want to get at the question of whether 1 is a prime number or not, then you ought to compare the prime numbers with 1, analyze their properties and their roles within the integers, then compare how these things extend or fail to extend when you move on to other mathematical structures, like polynomials and Gaussian integers. *This* is how you answer the question. Appealing to the history of mathematics actually reinforces most people's misconception that 1 should be considered a prime number, and reading the comments to this video has resoundingly confirmed this suspicion. I think that discussing the history is perfectly fine when addressing the question "why did we ever consider 1 a prime number?" or "how has our understanding of prime numbers changed?" But neither of those questions is the question the video claims to address.
I think it made more sense when I learnt about the ring theory way of thinking where you have units other than 1 (eg -1 for those wondering). Otherwise classifying 1 as it's own thing seems slightly hacky
Yeah, I'm surprised he didn't mention that, it's the most powerful argument against primality of 1 in my opinion P.S. Fun fact: in my first language "units" are called "one divisors" by analogy with "zero divisors" and it's such a nice name to explain the concept quickly IMO
@@vladislavanikin3398 It is the most powerful argument, and I would say, the only actual argument that really succeeds. All other arguments rely on logic that is fallacious, uncompelling. For example, this whole "it makes the fundamental theorem of arithmetic easier to state" nonsense is highly specious outside of the context of unique factorization domains. In most other areas of mathematics, mathematicians are completely unbothered when theorems have exceptions built into them. Heck, even for other theorems about prime numbers, there are many, *many* theorems which have 2, 3, and sometimes even 5 as the exception, and yet no one bats eye in calling these prime numbers anyway. The double standard makes no sense. Besides, the validity of a theorem should not depend so much on the precise details of how its phrased. Otherwise, you could prove literally anything by choosing "the adequate phrasing." This makes it obvious that the actual answer to the question "why is 1 not a prime number?" has nothing to do with the fundamental theorem. The question is answered, as you said, by the fact that 1 is a unit, and that units are conceptually and fundamentally different from irreducible elements.
Now I understand ! 1 is a prime number IF we work in the additive/Goldbach domain. It is not in the multiplicative domain because 1 is the neutral element of multiplication (which is most used because Goldbach's conjecture is not yet proved)
every natural number is a uniquely defined product of primes, unique up to ordering. the number one is the product of no primes, also known as the empty product.
An approach I heard in grade school is this: A prime number is a natural number with exactly two unique integer factors. 1 doesn't have two factors, only one.
@@forbidden-cyrillic-handle Or, you take the more reasonable approach of noting that all prime numbers have 4 divisors (-p, -1, 1, p), and 1 only has 2 divisors (-1, 1).
@@forbidden-cyrillic-handle Uh... I should probably remind you that -i and i *are not integers.* However, if you want to talk about *Gaussian integers,* then that is perfectly fine, then we can talk about -i and i as units alongside -1 and 1. In fact, the concept of a prime number is well-defined for all GCD domains. The integers, the Gaussian integers, and the ring of polynomials over the integers, are just examples of that.
@@forbidden-cyrillic-handle That is false. What I suggested is actually a definition used by mathematicians routinely today. The only person being unreasonable here is you, talking about pseudomathematical nonsense like numerology and what not.
14:56 "Every integer can be written uniquely as a product of primes" This was a simplification of what was highlighted in the text, but I'll address it anyways. That statement is false if one is not a prime, because one is an integer that would therefore not be able to be written as a product of primes. To address this, I would say a better elaboration of an integer would be "Every integer can be written uniquely as a minimal number of prime numbers". This way, it retains the fact that one is not *necessarily* included in the factorization of a number. This elaboration would define 1 as a prime number, which I'll get to in a bit. This would not have issue with the quotient formula, as you simply use the minimal number of prime numbers in the formula, which doesn't include one because its not needed and therefore not included in the minimum number of prime factors. It also works better with q(1) than the currently held definition, as the current trick to solving it would be to say q(1) = 1 * .... we ran into a problem didn't we? there is no prime factorization of one if one isn't a prime, so q(1) is not solvable using the trick. If instead we use my reasoning for integers and say one is a prime, we'd use the minimal number of prime factors of one, therefore we'd just use 1. q(1) = 1 * (1-1) = 1 * 0 = 0, there are no integers less than 1 that share a factor with it. This trick to solving the equation now still holds true, even if it was a trivial matter to do. Also, the p|ab argument I heavily disagree with. Just because something is trivially true, doesn't make it problematic, it just makes it not informative. But it is no more informative than say 3|6*9 since all that's saying is 3 divides at least one, but we know it divides both, just like 1 divides both a and b. So just because it isn't useful if it was defined as a prime, doesn't mean it shouldn't be a prime. By changing the definition, I made your erroneous simplification valid, I made the method for solving q(n) still hold true, and also solvable for q(1), and I've stated why p|ab is an inherently subjective reason for arguing against one being prime and therefore not a valid argument for a system that should be objectively true as much as possible. The only issue I can see is that there is a disconnect with the relationship between a number and its possible factors, as there is only one-way uniqueness now, meaning any set of factors points to exactly one integer, but one integer can be written as multiple factorizations. This is trivial in my opinion, as the wording we've used is "all prime factors other than one", when we could reinforce the definition by saying "the minimal number of prime factors". The only issue is saying it, because its not as succinct as "all prime factors", which again, subjective.
The saddest thing about Pluto is that the definition you cite actually *was* crafted to fit our _modern_ usage of 'planet'. For most of history planets were any large circular object - moons were called planets, for example. This usage was lost, briefly, and replaced by the planets of astrology. The planetary society than ham-fistedly shoved through the vote in the dying hours of a conference, and the rest is history.
All basis of modern mathematics don't consider 1 as prime, as for 0, its properties diverge from the ones of any other number (except 0, also having some weird properties, and sometimes considered as prime, depending on the context), and is the "base brick" for set theoriy, as S(0) [yes, 0 again, maybe 0 is the base brick for the theory of everything else ?]
13:06 I do think that if we consider one to be a number, and define primality as "the numbers that can only be measured by 1", it would still make sense to exclude 1 from the primes, because *1 can't even be measured by 1* (a number can not be measured by itself: 3:23)... Interesting math history video anyway!
Expert on video: if p|ab then p|a or p|b Me, an intellectual: if plab, then pla or plb As soon as I read that wrong, someone came through the window and took my degree away.
As a number theorist, I can tell you there are _countless_ reasons to exclude 1; so many I wouldn't even know where to begin. Excluding 1, if I tried to list all the theorems that need to be phrased "let p be a prime or 1," then I would be hard pressed to find even a single theorem. Including 1, if I tried to list all the theorems that would need to be phrased "let p be a prime other than 1," then I might have to list every theorem which refers to primes at all. Almost every pattern regarding prime numbers breaks at 1, suggesting it's more useful to exclude it.
I think that's pretty much the conclusion I reached in the video. I just wanted to streamline the examples I gave so as not to overwhelm viewers who weren't as familiar with the subject
@@AnotherRoof Yeah it’s definitely not the most intuitive fact, as evidenced by all the historical examples you gave. It only becomes obvious after seeing what we can do with primes.
I get where you're coming from, but I find the planet analogy a bit iffy, as it was decided without asking any planetary geologists. That said, I loved the video and I especially love your taste in movies.
Very Modern: Consider Gaussian primes. There are four "one"s and it's not negligible e.g. 2=i(1-i)^2. In such cases, one is considered to be in a special class separated from primarity.
I'm currently doing research with prime numbers. Interestingly, I sometimes find one fits as being prime, and sometimes it doesn't. I'm not convinced one way or another but it is easier to not think of one as prime.
Great video. I would like to see some similar subjects. For example, for modern French system 0 is a natural number and 0 is positive but not strictly positive. For modern american system, 0 is not a natural number, it is non-negative but it is not positive. (Personally I perfer without negation, namely to say positive/strictly positive rather than positive/non-negative, but just my subjective thought) Would like to know how historically these things are taken into considerations.
Using "Planet" here is kind of funny because of the original meaning of it. The ancient Greeks didn't have the same idea of what a planet was. The word means wonderer. They believed they were moving stars. People had no concept of the modern definition of "planet" until the 1600's when the telescope was invented. Interesting to see this come up in a video about how the classifications of 1 as a prime has also changed over time.
Since the definition of prime can vary, the only requirement is that any result referring to primes has to be consistent. So just because the same word is used two different ways doesn’t mean they are the same thing. A great example is the word “horse”. One use is a horse is a mammal with four legs, elongated head, hoofs, etc. But does a sea horse fit the definition. No. But it does share the property of elongated head. What about saw horse? It does share a property with the animal horse because it is used to carry a load. So if we used a different word it would clear up the problem. So let’s use the word “aaa” to refer to prime like numbers that are the set 3,5,7,11,… “eee” are the prime like numbers in the set 1,2,3,5,7,11,… and “iii” are the prime like numbers in the set 2,3,5,7,11,… Then the fundamental theorem of arithmetic could be stated using “iii” with no qualifications. To state it with “eee” you would have to add “except 1”. And so on. So the question is not so much, what is a prime? It is what is the easiest way to define a set of numbers that have certain properties, like divisibility, factors, and so on. The easiest way is to use one word. Again the main goal is consistency and your definition determines how other properties are stated. Now, is 0 a natural number????
The one thing I don't like about 1 not being a prime is that then it doesn't have a "prime signature" It's weird that 1 is impossible to construct using only prime numbers
Yeah this is an interesting point but because of 1's nature as the multiplicative identity there's really no escaping the fact that 1 is just weird and special and doesn't behave like any other number.
Great video. I wonder if the primality of 1 was the "Is Pluto a planet" pub discussion argument basis of its day. Of course we still have to contend with zero as an exclusion qualifier on so many functions. It could be a nice follow on to this video to do one on zero, especially if you get into varied number bases. Working title: [When did nothing become something?]
But ... if 1 isn't a prime then how can you write 1 as a sum of primes? In order for every integer to be written as a sum of primes 1 in and of itself must be prime
From the unique factorization point of view, primes have exactly one factor, composites have more than one factors and (the empty product) one has zero factors.
In my opinion, I don't consider 1 as a prime bc of the unique prime factorization of numbers, if 1 is prime, we have an infinite ways to express any number: 2 = 2*1 = 2*1² = 2*1³ and so on
@@MuffinsAPlenty I disagree. The truth is that 1 not being a prime number has absolutely nothing to do with the convenience of how we state the fundamental theorem of arithmetic. I dislike it when people who are educated in mathematics try to present everything as if it is "a matter of convenience," because that is just not true at all. 99.99999% of things in mathematics are never about convenience. Sure, _some_ things in mathematics are purely a matter of convention. The fact that we still use what I consider to be bad notation for derivatives, such as dy/dx, instead of using functional notation for them, such as D(y), really is entirely a matter of convention, even if there if one of the conventions is objectively superior to the other for three dozen reasons. It really is just notation. Using base 10 for our positional representation system to denote integers, instead of base, say, 60, like the Babylonians used to do, is s convention. In fact, using positional representations systems at all, rather than non-positional ones, is itself a convention. So, I am not saying there are no conventions in mathematics. My problem is that everyone on RUclips presents so many things (like 0! = 1, or 1 is not prime, or the ideas about the radical symbol being used as a function, etc.) as conventions that actually are not conventions. 1 not being a prime number is not a matter of notation, and it is not a choice we get to make. It is an irrefutable mathematical fact, that when it comes to commutative rings, and how we classify objects, there are exactly four families into which these objects can and do fall into, and these four families are exhaustive, distinct, and mutually exclusive. We can characterize these families as (a) those objects x such that there exists some y such that x•y = y•x = 0; (b) those objects x such that there exists some y such that x•y = y•x = 1; (c) those objects which are neither of the above, and whose proper divisors are exactly the objects in (b); (d) those objects which are not in (a) and have proper divisors in (c). How we choose to label these four families with four distinct labels is completely arbitrary, yes. We can even choose to have multiple distinct labels for the same individual family, yes. However, we can never choose to insist that elements from distinct families actually belong to one single family, and should be labeled as such. This is not a choice we get to make, because it is conceptually inconsistent with the mathematics above, and it leads to ill-defined terminology. Calling 1 a prime number is entirely analogous to insisting that my Toyota is a plant. Anyway, what this comes down to is, there is an actual conceptual reason behind why 1 is not and cannot be a prime number, no matter how much we would like it to be. It has nothing to do with how easy it is to formulate the language the factorization theorem in the English language when we reject 1 as a prime number. People should be taught the actual conceptual reason behind why 1 is a prime number, not this "it's more convenient" nonsense. And no, I am not saying we need to be formal about it. Simple intuitive explanations will do. I know that, as far as colloquial language is concerned, you can arbitrarily coin words and make them mean absolutely nothing and use them in self-contradicting fashion, or make them have useless meanings for the sake of trolling, and you can arbitrarily change how you use those labels any time you want to. But, this is not a colloquial language we are dealing with, now, or is it? We are dealing with abstract mathematics and number theory. It is a serious discipline of study. Going around telling biologists that your Toyota is a plant, because you chose to change the definition of the word "plant" in some unspecified, ad hoc way to include your Toyota in the definition, is not how science works. Similarly, simply changing the definition of terminology ad hoc so that 1 is a prime number, that is not mathematics. That is just pseudomathematical crankery. Now, I am not actually accusing anyone of having done this. That is not the point I am making. The point I am making is that educators need to stop encouraging this idea that all definitions exist only according to convenience, and that we change them willy-nilly how we want to. This is true of colloquial language, but not of language in academic disciplines of research. Educators also need to stop presenting fundamental mathematical facts that we do not get to do anything about as if they are something we choose. Perhaps you think otherwise, but that would beyond incomprehensible to me. Maybe I am out of my element. Maybe my strong advocation for the idea that people should not be taught false things makes me unreasonable, although if true, that gives me very little faith in the human species. But I remain unconvinced that this is the case.
@@angelmendez-rivera351 Thank you for your reply! Sorry it took so long to get back to you. I would say that, above, I went with a more "expedient" response than I usually would. Instead of going into my usual spiel about how one can notice that 1 (and other units) do not behave like primes exactly, and with careful enough definitions, we can pinpoint why, I chose to merely respond to the op granting the position that 1 was de-primed for convenience. I think my response was less of a defense of de-priming 1 for convenience sake and more of "whether or not I agree with your conclusion, I don't find your reasoning convincing" and was sloppy in my own execution of that. Now, I will say that I probably don't find saying 1 was de-primed out of convenience to be as bad of a thing as you probably find it to be. But I agree with your point that it isn't really actually the case.
No, not at all. There are many environments and interpretations that magnify just how much of a non-prime 1 is. The mu function value at 1 is +1, decidedly not -1 as would be required. This isn't just arbitrary assignment of value, either. Play around with the Legendre formula for the prime counting function to convince yourself that the [x/1] = [x] term in the sum needs to be positive, and could not at all be negative. In this context of a generalized Sieve of Eratosthenes, allow me to give my interpretation of 1. The primes are primes because they cast out all composites from your list of numbers from 1 to n. The number 1 writes down the list from 1 to n. This is essentially recognizing the pattern 1+1+1+...+1, 2+2+...+2, 3+3+...+3, 5+5+...+5, ... . The number 1 being added again and again is the table being made, and the primes being added is the actual Sieve in action upon that table. Since 1 can do nothing other than write the table down again (or erase the entire table as it goes, leaving neither primes nor composites, if that is your view), it is therefore decidedly and definitively not prime.
0:06 ok, here's my argument and we'll see how correct (or more likely how wrong) I am after the video: In my opinion 1 is not a prime number, because prime numbers need to have 2 factors with one of them being 1 and the other one being the prime number itself. 1 can't be prime, because it only has 1 factor. The factors aren't 1 and 1, that makes no sense, because you can't have multiple identical factors. And numbers need to have exactly 2 factors to be prime
Don't be so quick to dismiss what you see as nothing more than horoscope… nor should you underestimate the ancients' wisdom. The mathematical rigor of academic maths may be reconciled with the esoteric numerology on which people of as enduring intellect as Pythagoras, Plotinus, Iamblychus, and more have reasoned & written. I understand there is a lot of quackery, but a responsible sceptic doesn't dismiss that which he has not understood. "No empirical grounding according to 20th century understandings of the scientific method" is, ultimately, quite a finite criticism. Anyway I don't mean my only comment to be critical. I just shared on facebook that your pedagogical clarity & production quality with what I imagine is a shoestring budget is an inspiration for me. I love your videos.
It's too bad we don't actually teach the fundamental theorem of arithmetic in a fully qualified way... the most accurate way to say it is that each number, up to an invertible factor (1 and -1 for integers) has a decomposition into a unique product of primes, and some invertible factor. Said that way, it extends straightforwardly to more complex sets, such as the gaussian integers.
If you include -1 in your notion of "number," then you should include the negative primes -2, -3, etc as well. But then the product is no longer unique, only unique "up to associates" (ie if you have two factorizations, the primes of each can be paired up so that each prime in one product is an invertible factor times the corresponding prime in the other). And do you include 0 as a number? If so then what is the unique way to write 0 as a product of primes and an invertible element? Stating the full version precisely is actually very difficult to get right; there's way too much complexity for someone encountering the fundamental theorem of arithmetic for the first time to handle. I think it's much more reasonable to teach the version that restricts to positive integers first, and then at some point - after they've gotten comfortable with the idea but before seeing factorization in other rings - to ask them to think through and discuss how they would generalize the statement to work with the integers, without being able to refer to a notion of "positive." This would give them an opportunity to contend with the nuances of units, associates, zero divisors, etc before moving on.
@@japanada11 Yeah, I forgot to add that unique factorization only applies to multiplicatively cancellable numbers, which discounts zero. Thanks for pointing that out. I would take the view that -3 is a prime number, but it's "the same" prime as 3. Viewed that way, 6 = 2*3, 6 = (-2)*(-3), -6 = (-2)*3, and -6 = 2*(-3) are all "the same" factorization of "the same" number. (This is slightly different from what I said before, I know.) Explained by example like that, I don't think it's too hard to follow, though the precise abstract statement is indeed too complex to use as an introduction to the subject. I just think it's a shame that when we teach it, we don't even address the issue of negative numbers at all.
Fair point! Even if students are presented with the standard easier definition, I think it's important to get them to explore what happens with 1, with 0, with negative integers, with what happens if you allow fractions, etc. It really is a shame that unique factorization (as with so much in precollege math) is often presented as a fact to be learned and internalized, rather than a launchpad into many deeper questions!
It's not really accurate to say that Gauss was the first to prove the prime number theorem. The unique representation of numbers as products of prime factors was known even to Euclid. To understand why Euclid couldn't present a proof, look at his proof that there are infinitely many primes. It actually just proves that there are at least 4 primes. The point of the proof is that the same reasoning can be applied to any multitude of primes, and thus anyone who understands the proof can see that there are infinitely many primes. Similarly, Euclid's lemma (Elements VII Props. 30-32) can be applied repeatedly to eventually divide any number into prime factors. Euclid IX Prop 14 proves that this decomposition is unique. Many ancient authors clearly assumed this fact, so it was evidently known. Perhaps nobody ever thought to prove it, or had the notation to prove it in full generality, or perhaps those proofs are lost. But Gauss was merely writing down what people like Euler were already using in practice and had been for thousands of years.
I thought Eratosthenes, and his eponymous Sieve, were sufficient to show that 1 is not prime: “if a number is prime, eliminate all of its multiples and continue on until you find the next prime.” Since if you eliminate all multiples of 1, there are no numbers left, 1 cannot be prime.
I think that's putting the method before the definition. One could also use that line of reasoning to prove that 1 is the only prime, for example. Neat idea though and thanks for watching!
The problem is one is the first "number" so it should be prime. That is what prime means, like prime minister. All numbers are derived from 1, that is why they are divisible by one. Why not go back to the original idea and call the "prime numbers" incommensurate.
14:45 So if 1 is not prime, then 1 is not an integer? Maybe 1 is not a number? Maybe every number is 1 or part of 1? 1 is the Normal. It is not any particular length, but what you take it to be, and then all other numbers are normalised by 1. Is there a 0? Is it an integer? What primes make up 0? Maybe there is 0 (nothing) and 1 (all) - and everything else is between these limits? Maybe 1 is like the Unit Circle and 0 is its centre? Can all the infinities greater than 1 be mapped into the interval less than 1 and greater than 0? It seems like 1 is the Unity of all numbers. 1 is the Great Normal. A set or category is normalised by 1, and then can be counted 1, 2, 3 etc. Without a set or category, numbers have no meaning. There is nothing that is 2 or any other number compared to 1, if they are not part of a common set or category. Every individual thing is just 1, until there is a set or category.
13:08 - 13:19 It actually does not fit the definition. Earlier in the video, it was clarified that what mathematicians meant by "measured" was a notion of divisibility which only considersd proper divisors. In other words, the number of which the divisors are being considered does not measure itself, because it is not a proper divisor of itself, since it is not smaller than itself. This is included in the definition of a prime number. "A number is prime if and only if the only number that measures it is 1." Prime numbers are divisible by 1, AND by themselves, as well. This confirms that "measured by" and "divisible by" are not synonymous, because of the distinction between divisors and proper divisors. Prime numbers are divisible by themselves, but are not measured by themselves, they are only measured by 1. In modern terminology, what this means is that the only proper positive divisor of a prime number is 1. Now, it is clear that 1 does not fit the definition of a prime number: 1 is not measured by 1. It cannot be, because 1 is not a proper divisor of itself, by definition. In fact, 1 has no proper divisors. In the older terminology, this means that there no numbers that measure the number 1. Prime numbers are measured by the number 1, and only the number 1, but the number 1 is measured by no numbers at all. Therefore, it does not satisfy the definition given by medieval mathematicians of a prime number, even if you argue that 1 is actually a number at all. In other words, what this tells me is that medieval mathematicians were just inconsistent, and were incapable of detecting that treating 1 as a prime number was inconsistent with the definition of prime number they themselves used, since the definition of "measured by" that they used necessarily meant 1 was not measured by any numbers. They still came to the conclusion that 1 was measured by 1, because they were applying their own definitions inconsistently. This is something that was not uncommon back then, and it is the reason why rigor became a necessity later on.
It really just sounds like the answer is "It's more complicated than a binary answer." And let it be used where it should, and not used where it shouldn't.
18:58 well this isn’t wrong, it’s internally consistent. If one is prime, then every number shares a factor with twenty- one. So phi of n would be zero.
Every composite number n has at least one prime factor less than or equal to the square root of n. So you could say that for every natural number n > 1, if n does not have any prime factors less than or equal to sqrt(n), then n is prime. But you get a weird recursion if you apply this logic to n = 1: Suppose 1 is not prime. Therefore 1 has no prime factors
Not sure why 9 would be considered perfect... 6, on the other hand, does meet the definition of a perfect number: its non-self factors add to itself. A property shared by very few numbers. 28 and 496 are the next two smallest such numbers.
Well now that I think about it totient being always 0 if 1 is a prime totally makes sense, because every single number is divisible by 1 so all numbers share factor so there is exactly 0 of numbers that do not share factors
What's the square root of negative one? An infinite dimensional hypersphere of "imaginary" answers with all points of magnitude 1 All divisibility has a similar thing going on, positives use "split imaginary" numbers too
When I was at school - and I don't quite remember which degree of school it was - we did consider 1 to be one of primes, but many of our statements looked like "for all prime numbers greater than 1 ...". So yeah, 1 in my mind satisfies all logical requirements on primality but considering it a prime is impractical because then you have to exclude it way too often.
If you have to "exclude it too often," then it literally *does not* satisfy the conditions for primality. You would not have to exclude it so often if it actually did. Besides, if you take a look at the actual definition of a prime number, you will see that 1 is not one.
There is a concept called the empty product - it is a product with no factors. The result of taking a product with no factors is defined to be 1 (the multiplicative identity) because 1 is the only choice that would let the empty product be consistent with the associative property of multiplication. So in that sense, 1 is a product of 0 primes, and you _can_ get 1 using only prime factorizations, even if 1 isn't considered prime :)
So, then, would a fair definition of Prime that excludes one without doing so by fiat be 'Any number which has exactly two unique whole number factors'?
If you use the classic sieve approach of calculating primes, you start from 2 because if you start from one you eliminate every other number and then none of them are prime. Simply put, if one were a prime, it would be the ONLY prime.
Some observations - If we allow 1 to be considered a prime for the purpose of the totient function, then its result (always zero) is trivially also the correct answer to the question, as we would also need to allow 1 to be a permissible common factor. Since every integer has 1 as a common factor, there are always 0 integers which do not share a factor with any given integer. Thus, in this case, we get the necessity that "1 is not prime because 1 is not prime.", which is circular. Likewise, you state that the p|ab result is 'trivial' when p = 1, but being trivial does not make it false. You state that it does not 'tell us anything interesting', but I would counter that the information it contains is fundamental to the nature of integers - that they are all divisible by 1. Non-integers are not wholly divisible by 1, after all. While this result sounds trivial, it is only because it is so embedded into our understanding of integers that we feel it doesn't need stating; we are already restricting ourselves to looking at the integers, so including 1 gives no new information - 1 is trivially divisible because we are restricting ourselves to a regime where 1 is trivially divisible; circularity again. Further, there are a variety of results that require us to only view the odd primes for purposes of their result; this does not mean that 2 is *not* a prime for those purposes, only that those results specifically only hold for sets of primes (p >= 3). 2 is, itself, a strange and unique prime, which sometimes 'counts' and sometimes does not, depending on the need of the problem. The only difference between it and 1 in this sense is that more problems except 1 than except 2.
All fantastic points -- really enjoyed this comment! I'm pitching the video towards people who might not be as mathematically trained as us and I think these meta-mathematical points are a step more advanced than I wanted to go. But thanks for making them!
Watching this video, it's the first time I really understood why 1 isn't considered a Prime. I now think it makes sense. Before, I just was annoyed but accepted it because experts.
The number one should really be considered a planet.
I disagree - it deserves to be a dwarf planet
@@steampxnk2037 I was going to reply to say that exact same thing hah! You beat me to it!
One is obviously a vegetable
Would that Mercury ='s 1
Is 1 a sandwich?
I've always rationalised it even further by thinking that 1 is a sort of "superprime", which is sort of true because it is the multiplicative identity. We have to separate it from the normal primes because it screws with a lot of theorems involving prime numbers.
0 could be considered "supercomposite" since LCM(0,a) is a.
you are gay😂
Or you alter the theorems involving prime numbers. All of this is just made up anyway.
Primes become more and more common as you come down from infinity, culminating in infinite prime density at 1, or as you say, super-primeness
@X22GJP I disagree that it's all made up. I think alien civilizations discover the same numbers, and have the same debates.
I'm glad you're not strictly talking about math problems but also about the history and philosophy behind how we define things. And a video on deductive reasoning? Everything I ever wished for from a math channel.
Plenty on RUclips
I seem to remember 1 becoming "un-prime" when I was about 10. As you're classifying that as "medieval", I now feel positively ancient!
In the Contact movie/book it makes some sense to include '1' since the aliens were replying to early radio and TV emissions from 1936 when possibly the 1 was listed as a prime by some sources as you just shown.
Also, it makes sense to start a sequence with the fundamental unity of measurement in the same way that the
messages on the Voyager included distances based on a multiple (in base 2) of the fundamental frequency of the hydrogen atom.
@@forbidden-cyrillic-handle No. There is no way in which it is mathematically sound to actually consider 1 to be a prime number.
@@angelmendez-rivera351 I take it you didn't watch the video.
@@digitig Angel is a very well read mathematician, I'd say whatever Angel says overrides the video...
@@MagicGonads But this needs a historian of mathematics. If you read old mathematical papers, by the best mathematicians in history, you find that there was little agreement over whether to count 1 as a prone or not. There is agreement *now*, but as recently as my schooldays in the 60s, our textbooks still included 1 in the primes. And they weren't wrong, they were just on the verge of being rendered out of date.
@@digitig the historical context is largely irrelevant because the term "prime" is also irrelevant, it's the concept underlying that term that we need in order to address the question in the contemporary language of mathematics (the context in which the question is asked).
Just because historically "0" wasn't considered a number, does not mean the answer to the question "is 0 a number?" is "well we need to do a historical overview...", the answer is "it's an element of Z which are an archetypal example of a number set", the historical context may give you insight into why we currently consider 0 a number, but it won't decide the answer to the question.
But, there are actually matters of convention in dispute, like "does N (the natural numbers) contain 0?", "does the cup (subspace relation) include proper subspaces?", it does require extra clarification if you use the symbol in a formula where it might matter (but usually there's an expansion of the formula included in the relevant working/proof anyway that clarifies that instantly)
Some actually meaningful disputes are "most natural foundation / topos / calculus / axioms / construction?" which are things like 'intuitionistic' vs 'structuralist', that inform how you think about what underlies the mathematics you are doing (e.g. decides what standard of proof you stand by)
Contemporary mathematics is all about taking our disagreements as possible pathways to new mathematics, constraints and the lack of constraints both lead to new ideas. We would not be saying "1 is not prime because 1 is not a number", there would be a reason based on the properties of the underlying theory we have chosen to work with to decide this.
Another great video! I liked the bit about the Totient function as I hadn't come across it before!
I have come across it when determine size of various groups, such as U(20) would be 20 - 8 = 12 elements U(n) groups are groups when you pull out the numbers that are not relatively prime to n. And this group is a group under multiplication.
I love that you've kept up with the humor. It's really fun to watch your videos and I look forward to more!
21:33 that was BY FAR the most convincing argument i have heard for demoting pluto from planethood.
I feel like the totient function excluding 1 isn’t because 1 doesn’t work as a prime, but because it isn’t being counted as a factor. If you count all the numbers less than 20 that share a factor and you consider 1 to be a factor, then there are none because 1 is a factor of everything.
It works either way, if 1 is prime then totient of anything should be zero as the formula says
There's another definition of the totient function that is a little more structural and clearly doesn't depend on this choice. The totient of n counts all the numbers less than n which are units mod n. That is, it counts the numbers m < n for which there exists an associated integer d such that dm = 1 mod n, meaning that dm - 1 is divisible by n. This definition is clearly independent of whether you decide 1 is prime or not, and then including 1 as a prime totally messes up the count as shown in the video.
The reason 1 is not being counted as a factor is because it is not a prime number.
@@angelmendez-rivera351
I’ve seen you typing on a lot of these comments and I just have this to say:
If I say all green animals should be considered plants because they have the same color, the current definition of “plant” is irrelevant because we are discussing the definition itself.
Definitions are non-universal, made up by humans and always changing. Therefore you can discuss hypothetical scenarios were the current definition does not exist.
@@Alex-02 *If I say all green animals should be considered plants because they have the same color, the current definition of “plant” is irrelevant because we are discussing the definition itself.*
If you say all green animals should be considered plants because they have the same color, then this does not address the current definition of "plant" at all, and as such, it does not replace said definition. Also, you chose a poor example for an analogy, because not all plants are green. In any case, if you want to define categories of objects by their color, then all you have defined is the color itself. This has nothing to do with biology. You can even define a subclass of the category of living beings based on color. Again, though, this has nothing to do with biology: this is simply about the color of the living beings. You are still including green fungi, green protozoa, green bacteria, etc., all in this category. There is no biological property that green animals share with plants, such that no other entities share that property, living or non-living. Therefore, it is ontologically inadequate to insist that they are in the same category.
*Definitions are non-universal, made up by humans and always changing.*
Well, this is just a false belief. Definitions are not mere labels you come up with arbitrarily on the basis of preference for labeling objects of your preference. Definitions have to be consistent with reality. To apply definitions to a set of properties, you first have to ensure that these properties actually are well-defined, and that there exists at least one object which has those properties. Then, when you apply the label to this set of properties, it follows that the label applies to _all_ objects satisfying these properties. If you want to single out certain such objects as being qualified, and excluding the rest, then you need to specify properties which are satisfied exactly only by those objects you are trying to single out. Also, for this to be justified, you cannot do this on an ad hoc whim. Otherwise, that renders all definitions as completely redundant and useless. What makes definitions useful is precisely their non ad hoc nature. Finally, there is the problem of decidability. A definition has to be decidable, because if there exists no algorithm by which anyone would ever be able to tell if a class of objects satisfies the given definition or not, then it is completely pointless, and as such, the 'definition' does not actually define anything.
How precise you want to get with all of this, it all depends on what academic discipline of study you are a part of, or whether the definition is just intended for colloquial nonsense. In mathematics, though, the highest level of achievable precision with these definitions is required.
On the note of universality, mathematics actually are universal. Mathematics are independent of culture, nation, religious belief, etc. Yes, the types of notation you use to talk about mathematical concepts vary from language to language and culture to culture. The mathematical concepts themselves, though, are universal. A quasigroup is defined in Norway in exactly the same way it is defined in South Africa. A topological space is defined in Malaysia in exactly the same as it is defined in the United States of America. The works that you see published by mathematicians from New Zealand are not in disagreement regarding the concepts and their properties with the works published by mathematicians in China. You seem to insinuate that mathematical definitions are on the same caliber as definitions of words you find in the Urban Dictionary. Yes, those words do vary in definition from location to location, and even just from person to person, and really, those words are not defined in any particular way at all, they are used arbitrarily, because they are not used to discuss concepts, they are just used to communicate basic bits of information about a particular ill-defined thing. This is not how mathematical definitions work at all, though.
*Therefore you can discuss hypothetical scenarios were the current definition does not exist.*
I can guarantee you that there is no hypothetical scenario in which the current definition would not exist and in which people have the knowledge of ring theory that we do today.
Thank you for resolving a 50+ year annoyance! I learned some maths before formal schooling from some of my dad's 1930s textbooks: 1 was listed as a prime.
Then I went to secondary school and got (literally!) knuckle-rapped for answering 1 was prime, because the "new maths" had decided 2 was the first prime.
The discrepancy has always irritated me ever since. Your explanation from Gauss finally makes me happy!
Nice video: thank you.
I'm sorry the inconsistency got you into trouble but glad to have cleared it up!
Oh man, "until next prime" is a great closing stinger. If you used it just as a one-off for this episode, I urge you to consider making it your standard closing line.
the gap between primes increases very fast, we may never meet again
I think what I find so appealing about your videos is that they tend to take ideas that a lot of mathsy people are probably familiar with (the representation of numbers as sets, the idea that one is not a prime, etc.) and shine a light on why we got there, rather than just the nuts and bolts of how those ideas work. I hope this isn't accidentally an insult, but I feel like the most enticing part of your videos has been that they straddle the border between mathematics and philosophy, albeit clearly on team mathematics. I think there has been some exploration of this space, but usually not with the level of sincerity and depth that you bring. Once again, an exceptional video!
the still ongoing disagreement over the inclusion of 0 in the "natural numbers" is extremely annoying, i hope we solve it soon enough that we don't have to constantly use the unambiguous "positive integers" and "nonnegative integers" all the time
What convinced me that 1 isn't a prime was the logarithmic integral curve matching pi(n), that was like Nature telling us that 1 isn't a prime.
My favourite definition is "a number that has 2 unique factors" (no further clarification needed, provided that 1 and itself count as factors). 1 only has 1 unique factor, which means it has even less factors than a prime. Also the least number of factors of any integer (0 and infinity have infinity factors).
One way to justify the non-primality of 1 is to categorize natural numbers by their order in terms of divisibility. We say a | b ("a divides b") iff there is a natural number n so that an = b. Then we say a < b iff a | b and not b | a. Then you can verify that < is a strict partial order over N, and moreover, that 1 is the least element and 0 is the greatest element. That is to say, for any natural number n, 1 | n and n | 0. For any prime p, we have 1 < p, but there is no other number n for which n < p. So if you draw a graph of
Super interesting!
The different scripts for the 3 signs is a nice touch that I didn't notice until they were together at the end.
next topic will be about number theory, i see..
It's a bit like defining 0! = 1 - We just do it because the alternative is to have to keep track of the special case of 0 (or 1 for primes) at every stage of a proof.
If you're writing a proof you get to decide whether 1 is a prime number for the purpose of that proof. So many things were defined for just one proof and then the definition stuck
Another Roof, you have been selecting some awesome math topics! To create some connective tissue, I noticed the Euclidian definition of the unit and then the number as a collection of units sounds a lot like the foundation for the "empty set" definition of numbers you outlined in a previous video. I didn't realize that method of number definition had such old historical precedent...
Anyway, looking forward to future videos. Keep up the good work!
Until I started looking into it, I had no idea how long the "1 is its own thing and not a number" idea persisted!
My next video will be something very different.
@@AnotherRoof lol (x-1)! + 1 is divisible by x only if x is prime or 1
@@ValkyRiver Everything integer is divisible by 1.
I've been so looking forward to this, and you did not disappoint! Thank you for amazing math content that tickles my brain in a very wonderful way.
Another perfect video! Congratulations!
like the number 6
If it were perfect, it would be 6, 28 or (God forbid) 248 minutes long.
Great video. I really liked how you used this topic to talk about definitions and properties more broadly. I like how even if you’re talking about a ‘simpler’ topic you try to find the conclusion that helps a student to learn about math more broadly
Primes: *literally means first
Also primes: *excludes the first number
According to Peano, 0 is the first number anyway. 😛
@@irrelevant_noob still excludes it.
It's a cool video!!! Im hoping to see other topics in your channel cuase you explain so well but I enjoyed this one as well!
Many more videos to come my friend!
AAAAA i'm so glad you mentioned the distinction of primes/units in general rings like the integers! That's what I was watching for the whole time lol, definitely wish you mentioned a bit more about it but fantastic video as always!
I didn't want to go into too much detail on rings because they deserve their own video!
@@AnotherRoof I just binged your channel (+subscribed!) and from the Q&A you 100% seem like my kind of person, lol, so i have no doubt whenever you get around to it it'll be absolutely brilliant!
@@lexinwonderland5741 I agree with you that more should have been mentioned on it, since the distinction between irreducible elements and units in a ring is ultimately at the root of why 1 cannot be considered a prime number.
Look, do not get me wrong. I think that understanding how mathematical concepts from antiquity involved into mathematical concepts today as our understanding of mathematics improved and became more refined is very fascinating, and certainly an important kind of knowledge to have in general. However, as far as answering the question "is 1 a prime number?," the history is not enlightening at all: it ultimately does not answer the question. Yes, I know that mathematicians in the 1700s thought of 1 as a prime number, this is all well and fine, but that tells us nothing as to whether 1 actually is or should be considered a prime number or not. These questions are questions regarding the relationships between various mathematical concepts at a foundational level, not questions about names and conventions that mathematicians vote on. If you want to get at the question of whether 1 is a prime number or not, then you ought to compare the prime numbers with 1, analyze their properties and their roles within the integers, then compare how these things extend or fail to extend when you move on to other mathematical structures, like polynomials and Gaussian integers. *This* is how you answer the question.
Appealing to the history of mathematics actually reinforces most people's misconception that 1 should be considered a prime number, and reading the comments to this video has resoundingly confirmed this suspicion. I think that discussing the history is perfectly fine when addressing the question "why did we ever consider 1 a prime number?" or "how has our understanding of prime numbers changed?" But neither of those questions is the question the video claims to address.
@@angelmendez-rivera351 Do you know a good video or resource explaining this ring idea? (I don't know anything about rings).
I think it made more sense when I learnt about the ring theory way of thinking where you have units other than 1 (eg -1 for those wondering). Otherwise classifying 1 as it's own thing seems slightly hacky
Yeah, I'm surprised he didn't mention that, it's the most powerful argument against primality of 1 in my opinion
P.S. Fun fact: in my first language "units" are called "one divisors" by analogy with "zero divisors" and it's such a nice name to explain the concept quickly IMO
@@vladislavanikin3398 wow that actually makes a lot of sense re one-divisors
@@vladislavanikin3398 It is the most powerful argument, and I would say, the only actual argument that really succeeds. All other arguments rely on logic that is fallacious, uncompelling. For example, this whole "it makes the fundamental theorem of arithmetic easier to state" nonsense is highly specious outside of the context of unique factorization domains. In most other areas of mathematics, mathematicians are completely unbothered when theorems have exceptions built into them. Heck, even for other theorems about prime numbers, there are many, *many* theorems which have 2, 3, and sometimes even 5 as the exception, and yet no one bats eye in calling these prime numbers anyway. The double standard makes no sense. Besides, the validity of a theorem should not depend so much on the precise details of how its phrased. Otherwise, you could prove literally anything by choosing "the adequate phrasing." This makes it obvious that the actual answer to the question "why is 1 not a prime number?" has nothing to do with the fundamental theorem. The question is answered, as you said, by the fact that 1 is a unit, and that units are conceptually and fundamentally different from irreducible elements.
Now I understand !
1 is a prime number IF we work in the additive/Goldbach domain.
It is not in the multiplicative domain because 1 is the neutral element of multiplication (which is most used because Goldbach's conjecture is not yet proved)
Great video as always. I would really appreciate your aproach for the history of number 0.
every natural number is a uniquely defined product of primes, unique up to ordering. the number one is the product of no primes, also known as the empty product.
An approach I heard in grade school is this: A prime number is a natural number with exactly two unique integer factors.
1 doesn't have two factors, only one.
Not true. 1=(-1)(-1). You have to say 2 positive factors
@@forbidden-cyrillic-handle Or, you take the more reasonable approach of noting that all prime numbers have 4 divisors (-p, -1, 1, p), and 1 only has 2 divisors (-1, 1).
@@forbidden-cyrillic-handle Uh... I should probably remind you that -i and i *are not integers.* However, if you want to talk about *Gaussian integers,* then that is perfectly fine, then we can talk about -i and i as units alongside -1 and 1. In fact, the concept of a prime number is well-defined for all GCD domains. The integers, the Gaussian integers, and the ring of polynomials over the integers, are just examples of that.
@@forbidden-cyrillic-handle No, that is not the definition of "prime" in mathematics in the year 2023.
@@forbidden-cyrillic-handle That is false. What I suggested is actually a definition used by mathematicians routinely today. The only person being unreasonable here is you, talking about pseudomathematical nonsense like numerology and what not.
I enjoyed the history-centred approach :)
James Grime did a nice video on primes and the fundamental theorem of arithmetic over on numberphile, good companion to this video.
14:56 "Every integer can be written uniquely as a product of primes"
This was a simplification of what was highlighted in the text, but I'll address it anyways. That statement is false if one is not a prime, because one is an integer that would therefore not be able to be written as a product of primes.
To address this, I would say a better elaboration of an integer would be "Every integer can be written uniquely as a minimal number of prime numbers". This way, it retains the fact that one is not *necessarily* included in the factorization of a number. This elaboration would define 1 as a prime number, which I'll get to in a bit.
This would not have issue with the quotient formula, as you simply use the minimal number of prime numbers in the formula, which doesn't include one because its not needed and therefore not included in the minimum number of prime factors. It also works better with q(1) than the currently held definition, as the current trick to solving it would be to say q(1) = 1 * .... we ran into a problem didn't we? there is no prime factorization of one if one isn't a prime, so q(1) is not solvable using the trick. If instead we use my reasoning for integers and say one is a prime, we'd use the minimal number of prime factors of one, therefore we'd just use 1. q(1) = 1 * (1-1) = 1 * 0 = 0, there are no integers less than 1 that share a factor with it. This trick to solving the equation now still holds true, even if it was a trivial matter to do.
Also, the p|ab argument I heavily disagree with. Just because something is trivially true, doesn't make it problematic, it just makes it not informative. But it is no more informative than say 3|6*9 since all that's saying is 3 divides at least one, but we know it divides both, just like 1 divides both a and b. So just because it isn't useful if it was defined as a prime, doesn't mean it shouldn't be a prime.
By changing the definition, I made your erroneous simplification valid, I made the method for solving q(n) still hold true, and also solvable for q(1), and I've stated why p|ab is an inherently subjective reason for arguing against one being prime and therefore not a valid argument for a system that should be objectively true as much as possible. The only issue I can see is that there is a disconnect with the relationship between a number and its possible factors, as there is only one-way uniqueness now, meaning any set of factors points to exactly one integer, but one integer can be written as multiple factorizations. This is trivial in my opinion, as the wording we've used is "all prime factors other than one", when we could reinforce the definition by saying "the minimal number of prime factors". The only issue is saying it, because its not as succinct as "all prime factors", which again, subjective.
One is prime. I don't care if it messes up someone's arbitrary definition of some other thing. One is _the_ prime.
The saddest thing about Pluto is that the definition you cite actually *was* crafted to fit our _modern_ usage of 'planet'. For most of history planets were any large circular object - moons were called planets, for example. This usage was lost, briefly, and replaced by the planets of astrology. The planetary society than ham-fistedly shoved through the vote in the dying hours of a conference, and the rest is history.
Did Hector Salamanca just call me a numerologist?
Only if you discard scientific rigour.
No, he called you a *ding* *ding* *ding*
I’m glad you managed to squeeze at least one brick into the video at the end. I was getting worried.
All basis of modern mathematics don't consider 1 as prime, as for 0, its properties diverge from the ones of any other number (except 0, also having some weird properties, and sometimes considered as prime, depending on the context), and is the "base brick" for set theoriy, as S(0) [yes, 0 again, maybe 0 is the base brick for the theory of everything else ?]
I really enjoyed the historical walkthrough. Thanks for recommending this video during your charity marathon.
13:06 I do think that if we consider one to be a number, and define primality as "the numbers that can only be measured by 1", it would still make sense to exclude 1 from the primes, because *1 can't even be measured by 1* (a number can not be measured by itself: 3:23)... Interesting math history video anyway!
Expert on video: if p|ab then p|a or p|b
Me, an intellectual: if plab, then pla or plb
As soon as I read that wrong, someone came through the window and took my degree away.
As a number theorist, I can tell you there are _countless_ reasons to exclude 1; so many I wouldn't even know where to begin. Excluding 1, if I tried to list all the theorems that need to be phrased "let p be a prime or 1," then I would be hard pressed to find even a single theorem. Including 1, if I tried to list all the theorems that would need to be phrased "let p be a prime other than 1," then I might have to list every theorem which refers to primes at all. Almost every pattern regarding prime numbers breaks at 1, suggesting it's more useful to exclude it.
I think that's pretty much the conclusion I reached in the video. I just wanted to streamline the examples I gave so as not to overwhelm viewers who weren't as familiar with the subject
@@AnotherRoof Yeah it’s definitely not the most intuitive fact, as evidenced by all the historical examples you gave. It only becomes obvious after seeing what we can do with primes.
This is a really interesting video, and I say this as someone who didnt study maths beyond GCSE
I never knew Hector Salamance was into pure math
I’m surprised more of the comments aren’t about that “water is wet” premise
I get where you're coming from, but I find the planet analogy a bit iffy, as it was decided without asking any planetary geologists.
That said, I loved the video and I especially love your taste in movies.
Very Modern:
Consider Gaussian primes. There are four "one"s and it's not negligible e.g. 2=i(1-i)^2. In such cases, one is considered to be in a special class separated from primarity.
1 may or may not be a number, but 2 is the oddest prime.
Is it due to the fact it's the evenest one?
I'm currently doing research with prime numbers. Interestingly, I sometimes find one fits as being prime, and sometimes it doesn't. I'm not convinced one way or another but it is easier to not think of one as prime.
Absolutely brilliant series of videos. A thousand thanks for taking the time to create these. More as soon as possible, please!!! 😃
Great video. I would like to see some similar subjects. For example, for modern French system 0 is a natural number and 0 is positive but not strictly positive. For modern american system, 0 is not a natural number, it is non-negative but it is not positive.
(Personally I perfer without negation, namely to say positive/strictly positive rather than positive/non-negative, but just my subjective thought)
Would like to know how historically these things are taken into considerations.
Using "Planet" here is kind of funny because of the original meaning of it. The ancient Greeks didn't have the same idea of what a planet was. The word means wonderer. They believed they were moving stars. People had no concept of the modern definition of "planet" until the 1600's when the telescope was invented. Interesting to see this come up in a video about how the classifications of 1 as a prime has also changed over time.
Since the definition of prime can vary, the only requirement is that any result referring to primes has to be consistent. So just because the same word is used two different ways doesn’t mean they are the same thing. A great example is the word “horse”. One use is a horse is a mammal with four legs, elongated head, hoofs, etc. But does a sea horse fit the definition. No. But it does share the property of elongated head. What about saw horse? It does share a property with the animal horse because it is used to carry a load.
So if we used a different word it would clear up the problem. So let’s use the word “aaa” to refer to prime like numbers that are the set 3,5,7,11,… “eee” are the prime like numbers in the set 1,2,3,5,7,11,… and “iii” are the prime like numbers in the set 2,3,5,7,11,…
Then the fundamental theorem of arithmetic could be stated using “iii” with no qualifications. To state it with “eee” you would have to add “except 1”. And so on.
So the question is not so much, what is a prime? It is what is the easiest way to define a set of numbers that have certain properties, like divisibility, factors, and so on. The easiest way is to use one word.
Again the main goal is consistency and your definition determines how other properties are stated.
Now, is 0 a natural number????
The one thing I don't like about 1 not being a prime is that then it doesn't have a "prime signature"
It's weird that 1 is impossible to construct using only prime numbers
1 is a special number that is neither prime nor composite. It is similar to zero who is neither positive nor negative.
Doesn't it kind of have a prime signature, that being a completely empty set of primes?
Yeah this is an interesting point but because of 1's nature as the multiplicative identity there's really no escaping the fact that 1 is just weird and special and doesn't behave like any other number.
If you allow for empty products, then 1 is possible to construct using only prime numbers. 1 is the product of no prime numbers :P
@@trolledwoods377 Yes.
Great video. I wonder if the primality of 1 was the "Is Pluto a planet" pub discussion argument basis of its day.
Of course we still have to contend with zero as an exclusion qualifier on so many functions. It could be a nice follow on to this video to do one on zero, especially if you get into varied number bases. Working title: [When did nothing become something?]
But ... if 1 isn't a prime then how can you write 1 as a sum of primes? In order for every integer to be written as a sum of primes 1 in and of itself must be prime
I just love your style of explaining!
From the unique factorization point of view, primes have exactly one factor, composites have more than one factors and (the empty product) one has zero factors.
In my opinion, I don't consider 1 as a prime bc of the unique prime factorization of numbers, if 1 is prime, we have an infinite ways to express any number: 2 = 2*1 = 2*1² = 2*1³ and so on
2:23 all hail to the monad in the category of endofunctors.
The 'convenience' argument is always funny to me because of how many algebraic theorems state "For any nonzero number x..."
It's a balance. The convenience of zero existing far outweighs the inconvenience of excluding it from many theorems.
@@MuffinsAPlenty I disagree. The truth is that 1 not being a prime number has absolutely nothing to do with the convenience of how we state the fundamental theorem of arithmetic. I dislike it when people who are educated in mathematics try to present everything as if it is "a matter of convenience," because that is just not true at all. 99.99999% of things in mathematics are never about convenience. Sure, _some_ things in mathematics are purely a matter of convention. The fact that we still use what I consider to be bad notation for derivatives, such as dy/dx, instead of using functional notation for them, such as D(y), really is entirely a matter of convention, even if there if one of the conventions is objectively superior to the other for three dozen reasons. It really is just notation. Using base 10 for our positional representation system to denote integers, instead of base, say, 60, like the Babylonians used to do, is s convention. In fact, using positional representations systems at all, rather than non-positional ones, is itself a convention. So, I am not saying there are no conventions in mathematics. My problem is that everyone on RUclips presents so many things (like 0! = 1, or 1 is not prime, or the ideas about the radical symbol being used as a function, etc.) as conventions that actually are not conventions. 1 not being a prime number is not a matter of notation, and it is not a choice we get to make. It is an irrefutable mathematical fact, that when it comes to commutative rings, and how we classify objects, there are exactly four families into which these objects can and do fall into, and these four families are exhaustive, distinct, and mutually exclusive. We can characterize these families as (a) those objects x such that there exists some y such that x•y = y•x = 0; (b) those objects x such that there exists some y such that x•y = y•x = 1; (c) those objects which are neither of the above, and whose proper divisors are exactly the objects in (b); (d) those objects which are not in (a) and have proper divisors in (c). How we choose to label these four families with four distinct labels is completely arbitrary, yes. We can even choose to have multiple distinct labels for the same individual family, yes. However, we can never choose to insist that elements from distinct families actually belong to one single family, and should be labeled as such. This is not a choice we get to make, because it is conceptually inconsistent with the mathematics above, and it leads to ill-defined terminology. Calling 1 a prime number is entirely analogous to insisting that my Toyota is a plant.
Anyway, what this comes down to is, there is an actual conceptual reason behind why 1 is not and cannot be a prime number, no matter how much we would like it to be. It has nothing to do with how easy it is to formulate the language the factorization theorem in the English language when we reject 1 as a prime number. People should be taught the actual conceptual reason behind why 1 is a prime number, not this "it's more convenient" nonsense. And no, I am not saying we need to be formal about it. Simple intuitive explanations will do.
I know that, as far as colloquial language is concerned, you can arbitrarily coin words and make them mean absolutely nothing and use them in self-contradicting fashion, or make them have useless meanings for the sake of trolling, and you can arbitrarily change how you use those labels any time you want to. But, this is not a colloquial language we are dealing with, now, or is it? We are dealing with abstract mathematics and number theory. It is a serious discipline of study. Going around telling biologists that your Toyota is a plant, because you chose to change the definition of the word "plant" in some unspecified, ad hoc way to include your Toyota in the definition, is not how science works. Similarly, simply changing the definition of terminology ad hoc so that 1 is a prime number, that is not mathematics. That is just pseudomathematical crankery. Now, I am not actually accusing anyone of having done this. That is not the point I am making. The point I am making is that educators need to stop encouraging this idea that all definitions exist only according to convenience, and that we change them willy-nilly how we want to. This is true of colloquial language, but not of language in academic disciplines of research. Educators also need to stop presenting fundamental mathematical facts that we do not get to do anything about as if they are something we choose. Perhaps you think otherwise, but that would beyond incomprehensible to me. Maybe I am out of my element. Maybe my strong advocation for the idea that people should not be taught false things makes me unreasonable, although if true, that gives me very little faith in the human species. But I remain unconvinced that this is the case.
@@angelmendez-rivera351 Thank you for your reply! Sorry it took so long to get back to you. I would say that, above, I went with a more "expedient" response than I usually would. Instead of going into my usual spiel about how one can notice that 1 (and other units) do not behave like primes exactly, and with careful enough definitions, we can pinpoint why, I chose to merely respond to the op granting the position that 1 was de-primed for convenience. I think my response was less of a defense of de-priming 1 for convenience sake and more of "whether or not I agree with your conclusion, I don't find your reasoning convincing" and was sloppy in my own execution of that.
Now, I will say that I probably don't find saying 1 was de-primed out of convenience to be as bad of a thing as you probably find it to be. But I agree with your point that it isn't really actually the case.
أشكرك على المحتوى المتقن والجهد الكبير الذي تبذله في إعداد الفيديوهات
No, not at all. There are many environments and interpretations that magnify just how much of a non-prime 1 is. The mu function value at 1 is +1, decidedly not -1 as would be required. This isn't just arbitrary assignment of value, either. Play around with the Legendre formula for the prime counting function to convince yourself that the [x/1] = [x] term in the sum needs to be positive, and could not at all be negative.
In this context of a generalized Sieve of Eratosthenes, allow me to give my interpretation of 1. The primes are primes because they cast out all composites from your list of numbers from 1 to n. The number 1 writes down the list from 1 to n. This is essentially recognizing the pattern 1+1+1+...+1, 2+2+...+2, 3+3+...+3, 5+5+...+5, ... . The number 1 being added again and again is the table being made, and the primes being added is the actual Sieve in action upon that table. Since 1 can do nothing other than write the table down again (or erase the entire table as it goes, leaving neither primes nor composites, if that is your view), it is therefore decidedly and definitively not prime.
0:06 ok, here's my argument and we'll see how correct (or more likely how wrong) I am after the video:
In my opinion 1 is not a prime number, because prime numbers need to have 2 factors with one of them being 1 and the other one being the prime number itself. 1 can't be prime, because it only has 1 factor. The factors aren't 1 and 1, that makes no sense, because you can't have multiple identical factors. And numbers need to have exactly 2 factors to be prime
If I was an archaeologist I'd point out that talking about "hunter gatherers" as the pre antiquity form of society is... Antiquated
Don't be so quick to dismiss what you see as nothing more than horoscope… nor should you underestimate the ancients' wisdom. The mathematical rigor of academic maths may be reconciled with the esoteric numerology on which people of as enduring intellect as Pythagoras, Plotinus, Iamblychus, and more have reasoned & written.
I understand there is a lot of quackery, but a responsible sceptic doesn't dismiss that which he has not understood. "No empirical grounding according to 20th century understandings of the scientific method" is, ultimately, quite a finite criticism.
Anyway I don't mean my only comment to be critical. I just shared on facebook that your pedagogical clarity & production quality with what I imagine is a shoestring budget is an inspiration for me. I love your videos.
It's too bad we don't actually teach the fundamental theorem of arithmetic in a fully qualified way... the most accurate way to say it is that each number, up to an invertible factor (1 and -1 for integers) has a decomposition into a unique product of primes, and some invertible factor. Said that way, it extends straightforwardly to more complex sets, such as the gaussian integers.
If you include -1 in your notion of "number," then you should include the negative primes -2, -3, etc as well. But then the product is no longer unique, only unique "up to associates" (ie if you have two factorizations, the primes of each can be paired up so that each prime in one product is an invertible factor times the corresponding prime in the other). And do you include 0 as a number? If so then what is the unique way to write 0 as a product of primes and an invertible element?
Stating the full version precisely is actually very difficult to get right; there's way too much complexity for someone encountering the fundamental theorem of arithmetic for the first time to handle. I think it's much more reasonable to teach the version that restricts to positive integers first, and then at some point - after they've gotten comfortable with the idea but before seeing factorization in other rings - to ask them to think through and discuss how they would generalize the statement to work with the integers, without being able to refer to a notion of "positive." This would give them an opportunity to contend with the nuances of units, associates, zero divisors, etc before moving on.
@@japanada11 Yeah, I forgot to add that unique factorization only applies to multiplicatively cancellable numbers, which discounts zero. Thanks for pointing that out.
I would take the view that -3 is a prime number, but it's "the same" prime as 3. Viewed that way, 6 = 2*3, 6 = (-2)*(-3), -6 = (-2)*3, and -6 = 2*(-3) are all "the same" factorization of "the same" number. (This is slightly different from what I said before, I know.) Explained by example like that, I don't think it's too hard to follow, though the precise abstract statement is indeed too complex to use as an introduction to the subject. I just think it's a shame that when we teach it, we don't even address the issue of negative numbers at all.
Fair point! Even if students are presented with the standard easier definition, I think it's important to get them to explore what happens with 1, with 0, with negative integers, with what happens if you allow fractions, etc. It really is a shame that unique factorization (as with so much in precollege math) is often presented as a fact to be learned and internalized, rather than a launchpad into many deeper questions!
Thank you for making such great content!
It's not really accurate to say that Gauss was the first to prove the prime number theorem. The unique representation of numbers as products of prime factors was known even to Euclid. To understand why Euclid couldn't present a proof, look at his proof that there are infinitely many primes. It actually just proves that there are at least 4 primes. The point of the proof is that the same reasoning can be applied to any multitude of primes, and thus anyone who understands the proof can see that there are infinitely many primes.
Similarly, Euclid's lemma (Elements VII Props. 30-32) can be applied repeatedly to eventually divide any number into prime factors. Euclid IX Prop 14 proves that this decomposition is unique. Many ancient authors clearly assumed this fact, so it was evidently known. Perhaps nobody ever thought to prove it, or had the notation to prove it in full generality, or perhaps those proofs are lost. But Gauss was merely writing down what people like Euler were already using in practice and had been for thousands of years.
I thought Eratosthenes, and his eponymous Sieve, were sufficient to show that 1 is not prime: “if a number is prime, eliminate all of its multiples and continue on until you find the next prime.”
Since if you eliminate all multiples of 1, there are no numbers left, 1 cannot be prime.
I think that's putting the method before the definition.
One could also use that line of reasoning to prove that 1 is the only prime, for example.
Neat idea though and thanks for watching!
The problem is one is the first "number" so it should be prime. That is what prime means, like prime minister.
All numbers are derived from 1, that is why they are divisible by one. Why not go back to the original idea and call the "prime numbers" incommensurate.
14:45 So if 1 is not prime, then 1 is not an integer? Maybe 1 is not a number? Maybe every number is 1 or part of 1?
1 is the Normal. It is not any particular length, but what you take it to be, and then all other numbers are normalised by 1.
Is there a 0? Is it an integer? What primes make up 0?
Maybe there is 0 (nothing) and 1 (all) - and everything else is between these limits?
Maybe 1 is like the Unit Circle and 0 is its centre?
Can all the infinities greater than 1 be mapped into the interval less than 1 and greater than 0?
It seems like 1 is the Unity of all numbers.
1 is the Great Normal. A set or category is normalised by 1, and then can be counted 1, 2, 3 etc.
Without a set or category, numbers have no meaning.
There is nothing that is 2 or any other number compared to 1, if they are not part of a common set or category.
Every individual thing is just 1, until there is a set or category.
I watched Contact a few weeks ago and I'm sure the prime number broadcast started at 2 in the film. I wonder why it was changed from the book.
13:08 - 13:19 It actually does not fit the definition. Earlier in the video, it was clarified that what mathematicians meant by "measured" was a notion of divisibility which only considersd proper divisors. In other words, the number of which the divisors are being considered does not measure itself, because it is not a proper divisor of itself, since it is not smaller than itself. This is included in the definition of a prime number. "A number is prime if and only if the only number that measures it is 1." Prime numbers are divisible by 1, AND by themselves, as well. This confirms that "measured by" and "divisible by" are not synonymous, because of the distinction between divisors and proper divisors. Prime numbers are divisible by themselves, but are not measured by themselves, they are only measured by 1. In modern terminology, what this means is that the only proper positive divisor of a prime number is 1. Now, it is clear that 1 does not fit the definition of a prime number: 1 is not measured by 1. It cannot be, because 1 is not a proper divisor of itself, by definition. In fact, 1 has no proper divisors. In the older terminology, this means that there no numbers that measure the number 1. Prime numbers are measured by the number 1, and only the number 1, but the number 1 is measured by no numbers at all. Therefore, it does not satisfy the definition given by medieval mathematicians of a prime number, even if you argue that 1 is actually a number at all. In other words, what this tells me is that medieval mathematicians were just inconsistent, and were incapable of detecting that treating 1 as a prime number was inconsistent with the definition of prime number they themselves used, since the definition of "measured by" that they used necessarily meant 1 was not measured by any numbers. They still came to the conclusion that 1 was measured by 1, because they were applying their own definitions inconsistently. This is something that was not uncommon back then, and it is the reason why rigor became a necessity later on.
Excelent video mate! Very useful for teaching purposes
It really just sounds like the answer is "It's more complicated than a binary answer." And let it be used where it should, and not used where it shouldn't.
This is a great video! For some reason, I had a really strong sense of déjà vu while watching it
maybe because my recording setup hasn't changed since my first video and won't be changing any time soon? :P
Until Next prime could become your outro text, it was amazing!
18:58 well this isn’t wrong, it’s internally consistent. If one is prime, then every number shares a factor with twenty- one. So phi of n would be zero.
This channel is AWESOME. Thank you
11:06
Water is makes things wet, not wet itself.
This is so satisfying. Thank you so much.
Every composite number n has at least one prime factor less than or equal to the square root of n. So you could say that for every natural number n > 1, if n does not have any prime factors less than or equal to sqrt(n), then n is prime.
But you get a weird recursion if you apply this logic to n = 1:
Suppose 1 is not prime.
Therefore 1 has no prime factors
Thank you for another one. Can't wait for me.
Interesting historical bent. Wasn't expecting that, but I enjoyed it all the same.
Not sure why 9 would be considered perfect...
6, on the other hand, does meet the definition of a perfect number: its non-self factors add to itself. A property shared by very few numbers. 28 and 496 are the next two smallest such numbers.
Well now that I think about it totient being always 0 if 1 is a prime totally makes sense, because every single number is divisible by 1 so all numbers share factor so there is exactly 0 of numbers that do not share factors
06:21 Live long and prosper.
What's the square root of negative one?
An infinite dimensional hypersphere of "imaginary" answers with all points of magnitude 1
All divisibility has a similar thing going on, positives use "split imaginary" numbers too
When I was at school - and I don't quite remember which degree of school it was - we did consider 1 to be one of primes, but many of our statements looked like "for all prime numbers greater than 1 ...". So yeah, 1 in my mind satisfies all logical requirements on primality but considering it a prime is impractical because then you have to exclude it way too often.
If you have to "exclude it too often," then it literally *does not* satisfy the conditions for primality. You would not have to exclude it so often if it actually did. Besides, if you take a look at the actual definition of a prime number, you will see that 1 is not one.
@@angelmendez-rivera351 why are you spamming the comment section?
@@TykoBrian7 I am not. Please refrain from making baseless accusations. Such behavior does make a good look for you.
7:11 is a fun book to read!
1 is technically prime because if you times 1 by itself, you're just get 1 and you can't multiply by any others numbers to get 1.
There is a concept called the empty product - it is a product with no factors. The result of taking a product with no factors is defined to be 1 (the multiplicative identity) because 1 is the only choice that would let the empty product be consistent with the associative property of multiplication.
So in that sense, 1 is a product of 0 primes, and you _can_ get 1 using only prime factorizations, even if 1 isn't considered prime :)
So, then, would a fair definition of Prime that excludes one without doing so by fiat be 'Any number which has exactly two unique whole number factors'?
This was an amazingly well made video!
If you use the classic sieve approach of calculating primes, you start from 2 because if you start from one you eliminate every other number and then none of them are prime. Simply put, if one were a prime, it would be the ONLY prime.
20:22 exactly definitions are use for humans. If opaque it becomes not useful and non definable...
In modern language, where artificially intelligence gains more and more importance this technique which you described here is called “fine-tuning”
You are the math hero we always wanted but never deserved ❤
Some observations - If we allow 1 to be considered a prime for the purpose of the totient function, then its result (always zero) is trivially also the correct answer to the question, as we would also need to allow 1 to be a permissible common factor. Since every integer has 1 as a common factor, there are always 0 integers which do not share a factor with any given integer. Thus, in this case, we get the necessity that "1 is not prime because 1 is not prime.", which is circular.
Likewise, you state that the p|ab result is 'trivial' when p = 1, but being trivial does not make it false. You state that it does not 'tell us anything interesting', but I would counter that the information it contains is fundamental to the nature of integers - that they are all divisible by 1. Non-integers are not wholly divisible by 1, after all. While this result sounds trivial, it is only because it is so embedded into our understanding of integers that we feel it doesn't need stating; we are already restricting ourselves to looking at the integers, so including 1 gives no new information - 1 is trivially divisible because we are restricting ourselves to a regime where 1 is trivially divisible; circularity again.
Further, there are a variety of results that require us to only view the odd primes for purposes of their result; this does not mean that 2 is *not* a prime for those purposes, only that those results specifically only hold for sets of primes (p >= 3). 2 is, itself, a strange and unique prime, which sometimes 'counts' and sometimes does not, depending on the need of the problem. The only difference between it and 1 in this sense is that more problems except 1 than except 2.
All fantastic points -- really enjoyed this comment! I'm pitching the video towards people who might not be as mathematically trained as us and I think these meta-mathematical points are a step more advanced than I wanted to go. But thanks for making them!
What are your thoughts on considering 0 to be a prime? 0|ab → (0|a or 0|b) is also trivially true (in the natural numbers, anyway).
Watching this video, it's the first time I really understood why 1 isn't considered a Prime. I now think it makes sense. Before, I just was annoyed but accepted it because experts.
Babylon had systematic math too, and it's older than Greece.