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?

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.

Plenty on KZfaq

So there is an option that we do not send 1 as a prime to the aliens and they may think we're awful and destroy us. Or vice versa.

@@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 And they weren't wrong, they were just on the verge of being rendered out of date.

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'm sorry the inconsistency got you into trouble but glad to have cleared it up!

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.

the gap between primes increases very fast, we may never meet again

Many more videos to come my friend!

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.

Only if you discard scientific rigour.

No, he called you a *ding* *ding* *ding*

like the number 6

If it were perfect, it would be 6, 28 or (God forbid) 248 minutes long.

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).

According to Peano, 0 is the first number anyway. 😛

​@@irrelevant_noob still excludes it.

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.

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

maybe because my recording setup hasn't changed since my first video and won't be changing any time soon? :P

Is it due to the fact it's the evenest one?

Not true. 1=(-1)(-1). You have to say 2 positive factors

@@victoramezcua4713 -1 is there twice. Two of them are not unique factors. Anyway he could have used the natural numbers again as factors instead of integers.

@@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).

@@angelmendez-rivera351 if I f we go that route then let's add also i and -i. Let's also add rational numbers. And why we study numerology at all? Obviously the initial definition of what is a prime number is numerology. Therefore let it burn.

@@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.

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 KZfaq 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.

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.

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.

The concept of a 'prime' number makes no sense in an additive sense. It is inherently and necessarily a multiplicative concept.

@@angelmendez-rivera351 I meant additive sense as repeated addition

@@1ToTheInfinity That does not address anything I have pointed out.

@@angelmendez-rivera351 seeing 42 as 7+7+7+7+7+7 or as a rectangle prioritizes the *pairs* of factors that multiply to 42, and since 1, 5, and 79 both can only be divided by 1 and itself, it makes perfect sense 1 is among the primes, but in a modern factorization sense of 2*3*7, 1 acts differently than 2,3 or 7 now that it is part of composing numbers, in which 1 doesn’t help with the prime factorisation

@@1ToTheInfinity *...since 1, 5, and 79, are all examples of positive integers which can only be divided by 1 and itself, it makes perfect sense 1 is among the primes...* No, it does not make sense. The problem is, "only divisible by 1 and itself" is not the definition of a prime number. It never really has been the definition of a prime number. The video talked about how, in the past, different definitions of prime number were used, and the one which was used the most was this notion of being "measured" by another number. For example, "the prime numbers are numbers that are measured only by the number 1 and nothing else." This is the definition that you encounter Another Roof mentioning throughout the video, because that is the definition that was used in antiquity and in medieval times. But saying that a number is divisible by x is _not_ the same as saying the number is measured by x. This was explicitly stated in the video too. Why are they not the same thing? Well, because (a) the mathematicians of those times did not consider numbers to be measured by themselves. In other words, 1 measures 7, but 7 does not measure 7. Again, this was explicitly stated in the video. The other reason is that (b), well, it simply is not true that 5 is divisible by only 5 and 1. 5 is also divisibly by -5 and -1. But for a long time, negative integers were never considered in number theory. This is an oversight that needs repair. Let me go back to part (a). As mentioned, 1 measures 7, but 7 does not measure 7. That is how mathematicians used to think of it, but why? What is the difference between saying that 7 measures 7, and that 7 divides 7? The difference is that 1 is a proper divisor of 7, but 7, although it is a divisor of 7, is not a _proper_ divisor of 7. The distinction between a divisor of x and a proper divisor of x is actually the exact distinction as the distinction between subset and proper subset. It is also completely analogous to the distinction between "equal or less than" and "less than." y is called a divisor of x if and only if y divides x. But, a proper divisor is more special. y is called a proper divisor of x if and only if y divides x AND x does not divide y. Now we get it: 1 divides 7, but 7 does not divide 1, so 1 is a proper divisor of 7. The whole point of proper divisors is that we only care about the divisors of x that are "simpler" than x: we do not at all care about the fact that x divides itself, because that is just completely useless, and trivial. Like, all numbers divide themselves anyway, so do we care about x being a divisor of x? This is why the concept of proper divisors exists. And the concept of proper divisors is the concept that the mathematicians of old were alluding to when they said "x measures y." The old language of "x measures y" translates into the modern language as "x is a proper positive divisor of y." Positive, because again, negative numbers were never considered seriously by European mathematicians prior to like the 1500s. So, now that you know what the distinction is between divisors and proper divisors, and now that you have watched the video and you understand that prime numbers were always ultimately defined in terms of proper divisors, albeit in a different language, it should be clear why 1 is not a prime number. You see, the definition of a prime number always has been "a positive number which is only measured (positively) by the number 1." Translating this to the modern language, this means "p is prime if its only positive proper divisor is the integer 1." _THIS_ is the definition of a prime number. This is what it has been since basically forever, in concept, even if the language used was different. But look: the number 1 has no positive proper divisors, since its only positive divisor 1 itself. So, it does not actually satisfy the definition of a prime number. The problem here is that most teachers, and most textbook authors, believe that the actual definition is too complicated for grade schoolers (i.e, children) to learn. So they simplify it down, they get rid of all the "technical details," and so they just tell the children that 'a prime number is divisible only by itself and by 1.' But what this is a mistake, because this is completely misleading: you have changed the definition itself altogether by getting rid of the technical details. You *cannot* get rid of the technical details, because they are the _most_ important part of the definition, and not the _least_ important part. Prime numbers always have exactly 4 divisors: -p, -1, 1, p. The proper divisors are -1 and 1. But -1 and 1 have no proper divisors at all! They are fundamentally different from the prime numbers, and do not satisfy the definition of a prime number, so they belong to an entirely different classification system. -1 and 1 are called "units," or unitary numbers. One of the defining properties of units is that they divide _all_ numbers. Also, the product of two units is always a unit. Notice how this can never be true with prime numbers: the product of two prime numbers necessarily is a composite number, by definition. Also, units cannot be divided by any prime numbers at all. They can only be divided by units. 2 cannot divide 1 or -1. -7 cannot divide 1 or -1. Units are fundamentally different from prime numbers. Saying 1 is a prime number is like saying "a car is an animal." Like, no, that is just way off. Also, 0 is not a prime number either. It is also not a unit, because you cannot divide by 0. 0 is what is called a zero divisor. In the integers, 0 is the only zero divisor, but this is not true for all systems of arithmetic, actually. For example, when you work with matrices, there are some matrices not equal to the zero matrix, but are zero divisors anyway. A zero divisor is a quantity x such that x•y = 0 for some nonzero y. By the way, this distinction between primes, units, and zero divisors, is not exclusive to the integers. It holds universally. It holds for matrices, polynomials, associative vector algebras, the rational numbers, the complex numbers, the Gaussian integers, the dual-integers, the split-complex integers, etc. It holds for all commutative systems with associative multiplication and addition. A unit is some quantity x such that for some y, x•y = 1. The integers -1 and 1 are units, and they are the only units in the integers. In the Gaussian integers, the units are 1, -1, i, -i. In the rational numbers, everything is a unit, except 0, which is a (trivial) zero divisor. The analogue for prime numbers in more general settings is called an irreducible element. An irreducible element is an element that (a) is not a zero divisor, (b) its only _proper_ divisors are units. Remember: units have no proper divisors, so they are not irreducible elements. A composite element is just a product of two or more irreducible elements, just like in the integers. Note: how you factorize composites into irreducibles need not be unique in general. But, in the integers, it is unique. Now that you know all this, you are probably going to object that this all just "the multiplicative perspective," and that when you look at it from "the additive perspective," it is very different. But that is not the case at all. Why? Because the definition of a prime number has *absolutely nothing* to do with how you write numbers as repeated sums of integers. In fact, there is nothing interesting to point out: all integers can be written as sums of two integers in infinitely many ways, and all nonzero integers can be written as sums of 1 or -1 alone. So, ultimately, the additive structure does not matter at all. My point is, there is no such a thing as "the additive perspective." Insisting that there is one is borne from a severe misunderstanding of how number theory actually works.

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!

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.

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!

I missed a trick there -- an infinite loop of videos would really rack up the views!

The most intuitive way I know of (which, for square roots, is equivalent to Newton's method) to get the square root of a number k is to first guess a number r_0. This is probably not the root, but you can find the other factor, k / r_0. Since you want to find a number r such that r = k / r, it makes sense to take the average of these two numbers and to try r_1 = (r_0 + k / r_0) / 2. This is a better estimate than r_0 of the true square root of k. Repeat as necessary. For root 2, suppose 1 simply start with a very dumb guess of r_0 = 1. Then r_1 = (1 + 2) / 2 = 3/2. r_2 = 17/12, and r_3 = 577/408. This is already correct to 5 decimal places.