 In the previous video we learned about the Euclidean algorithm and how we can use it to compute the GCDs of two integers and more importantly how we can write the GCD as a linear combination of the two original integers. And so I want to mention in this video some consequences of the Euclidean algorithm which will actually prove to us to be very useful. So for example if A and B are relatively prime that means their GCD is one that means there's some way of combining the two numbers together A, R plus B, S so that it's equal to one. There's some linear combination that equals one that's an immediate consequence of the theorem we saw earlier due to the Euclidean algorithm. Let's look at some other things. As a reminder what do we mean by a prime number right? Suppose that P is some number some integer greater than one. We say it's a prime number if the only positive factorizations of the number are P in itself right? The only way you can factor numbers is P times one. That's always a factorization of a number but for prime numbers these are the only positive factorizations of it. We say that a number that if it's not prime then we call it a composite number. And so when it comes to primes I mean even Euclide many thousand years ago understood the importance of prime numbers. And so the following statement is often referred to as Euclid's lima. If we have two numbers A and B, so these two integers positive or negative, and we have some prime number P, if P divides the product A times B then it must be that P divides A or P divides B. Now I should mention that this is very important that we have prime numbers in this situation. So for example if you took like A to B 3 and B to B 4 then notice that the product A times B is equal to 12. And if you take like oh 6 divides 12 because it's just 2 times 6. The problem is 6 does not divide, it doesn't divide 3 nor does 6 divide 4. So this is a potential counter example but oh it's a composite number 6 a composite number. And that's because A and B can kind of share when you put the product together they could have different primes you know or something like that. And therefore it is important that it's prime numbers in this consideration because the prime numbers in terms of number theory are these atomic Billy blocks. You can't factor something more than the prime factorization. And so let's see very quickly how one could prove Euclid's lima. So when you have like an either or statement in logic this is what you often call a disjunctive syllogism that is the proof technique we're about to use which if you have this or that what we can say is the following if it's not this then it's definitely that because it's kind of like two options right as a statement P divided by A is either true or it's false. That's the dichotomy we have when it comes to statements it's either true or false. So okay if it's true right if your statement A is true that then implies A or B right and so if the first statement is true then we're done. So in the situation that the statement is actually false we then have to show that the other statement is true. And so that's how one shows an or statement oftentimes which you'll take one of the statements and negate it and improve the other one has to be true. If it's either or not this then that's right. So just a little bit of a logical argument there this disjunctive syllogism. So we're going to assume that P does not divide A because if P did divide A we'd be done. So we don't have to consider that case because it's just too easy. So if P doesn't divide A that would actually suggest that the greatest common divisor between P and A is equal to one. This is significant and this follows from the fact that P is a prime number. The only divisors of P are one and P because it's a prime number. And so if P doesn't divide A then the GCD has to be one. Now since the numbers are then co-prime by the Euclidean algorithm we get that there are going to be numbers R and S such that AR plus PS equals one. All right. Let's multiply both sides of the equation by B. If we do that we get the following B times one of course is equal to B. AR times B is going to be ABR and PS times B will just be PS no big deal there. But then notice in terms of divisibility here right. So we have that P divides itself. We also have by assumption that P divides AB and therefore since P divides AB and P that would then force that B has to be divisible by P thus giving what we wanted. If P doesn't divide A then it must divide B instead. That's because it have a prime number. Where do we use the fact it was a prime number? The fact the number was prime forces the GCD here to be one for which then we use the linear combination. So this is quite interesting here. We only use this linear combination principle to show the divisibility here. We don't actually need to know what the numbers R and SR are in this situation. This is just a purely theoretical basis. The Euclidean algorithm does actually provide a way to compute R and SR if we need them. But oftentimes in proofs we don't actually need the exact numbers. We just need existence. Existence is sufficient. Euclid's lemma is a very powerful result and I want to show you and again Euclid's lemma gets its name because this result plus the Euclidean algorithm were placed in Euclid's the elements and one one powerful application of Euclid's lemma was that Euclid proved that there are infinitely many prime numbers. There's the set of prime numbers is not a finite set, which is extremely impressive. When you think of the time frame where Euclid did this work, the idea of infinity was a very, very, very well miss. It wasn't very, it was a misunderstood creature at the time, right? notions of calculus were were hundreds of not thousands of years away, right? Prototypes of calculus maybe a little bit closer, but it was millennia after Euclid's writing that we truly got a better understanding of what infinity means. But he was able to prove that there was no finite list of prime numbers, right? And there actually are many proofs of the infinitude of prime numbers. But as as least as far as I'm aware of in history, I think Euclid's is the first documented proof of the infinitude of prime numbers. And it will be a very quick consequence of Euclid's lemma, which we're going to see right here. This is going to be a classic proof of by contradiction. So we're going to suppose the contrary that there's only finitely many prime numbers. So all of the prime numbers in the world are P1, P2, up to PR. Then we're going to construct a number Q, which is the product of all finite number of all prime numbers, and then we're going to add one to it. Well, this number is clearly bigger than every prime number. And as such, it must be composite because there's there's we've exhausted all the possibilities bigger than every prime number. So it has to be composite, which means it's divisible by some prime number. And we're going to call that prime number PI right here. It's going to be divisible by one of them. And so this is where this is where Euclid's Euclid's lemma's going to come into play here, right? So take the number, take the number P right here. So this is this is a prime, right? It divides here and this should probably be PI. Sorry, that might be a typo right there. So PI is going to divide this because we know for a fact that PI divides Q by assumption, right? And we also know that PI divides the product of all the primes because PI is one of those there. So it's going to have to divide their difference, which is here is one. So PI divides one, that's a contradiction. So that would suggest that there is no prime divisor of Q. I mean, I guess Q itself could be a prime, but that's a problem, right? We can't we can't get that. And so we get we get here that Q must actually we get the contradictions and the contradiction being on the on the number of primes, there has to always be a bigger prime. And since this argument also gives us way of constructing numbers to be automatically relatively prime, if you had any collection of primes or any collection of numbers, we can always build a number to be relatively primed at them, just by taking the product of all of them and adding one to them. Alright, so this this infinitude of prime is a classic proof, which didn't follows by these divisibility arguments. Another thing we should mention be the fun well theorem of arithmetic, that every number has a unique factorization, every integer has unique factorization wouldn't mean we that's the following, take a positive integer. We first of all claim that it has some factorization. There are some primes that are going to divide it. Basically, for the reasons one can argue before here, this this comes from the well order in principle, that if in itself is not prime, right? Because I mean, if it's prime, that is your factorization, if it's not prime, then n will equal a times b where a and b are smaller numbers. And then by induction, you assume by your inductive hypothesis that Oh, this has a prime factorization, this has a prime factorization viola, you now have in has a has a factorization that comes from the induction axiom. What about uniqueness of factorization? This is where Euclid's limit comes into play. So for uniqueness of factorization, let's say you have two factorizations. Well, Pete say, for example, P one divides n, and therefore divides this thing right here. So P one must divide one of the factors of this thing right here. Now, each as each factor here is itself a prime. The only things that divide a prime are one or the number itself. So that's going to force that P one is equal to one of the q i's for which we often say, oh, let's just say without the loss of jelly, it's q one, then you divide that from consideration and then you invoke induction again at that moment, in which case this will show you that one to one, there's going to be this correspondence between the primes p one equals, you know, one p one equals q one p two equals q two, you know, up to reordering, right, we could call them that. And also the number of primes is going to be the same as well. So when you combine the induction axiom with Euclid's lima, you get the fundamental theorem of arithmetic. And so Euclid's lima is a very powerful tool when one comes to studying divisibilities. And it places huge importance on places huge importance on what prime numbers are. And this is sort of an interesting to bring up here because you'll notice that prime numbers were defined as those numbers with basically cannot be factored. But then Euclid's lima gives us this other tool about divisibility with respect to prime numbers. And later on, when we talk about rings, we will be interested in the future. This this is actually something that we talk about in the sequel math 4230, we talked about factorization in integral domains, which is a type of rings in terms that'll make much more sense in the future. But it turns out that there's two notions kind of going on here, the notion of a prime and the notion of irreducible elements. The way we define a prime number is typically what we would call an irreducible element in a general ring. And what we call a prime element is actually the things that have the property of Euclid's of Euclid's lima. And it just so happens that in the ring of integers, primes and irreducibility are the exact same thing. But in the future, we will explore situations where rings can actually have a separation of these two ideas irreducibles and primes can be a different set. And as such one, then we'll ask questions about when can we recapture the phonolatheum of arithmetic? This will be, for example, in a unique factorization domain. And so abstract algebra is all about taking these algebraic properties of the number systems we know and love, and we try to find them in some type of alien situation, right? You know, if we are traveling to Mars or some other, some other solar system, right, we find planets, we find life there, if you find life on the moon Titan, or whatever on Io or somewhere else in the cosmos, how how in the world is a biologist on earth going to be all studied life on such a planet on such a moon, right? Clearly, if life exists there, it could be very different, right? But how are you even going to know it's life? We're going to look for similarities between life on earth, and this extraterrestrial life. So there's going to be some comparisons that can be made. But also, there's going to be things that could be different. And so if you're going to be some type of like astrobiologist, maybe that's going to be a thing someday. Then you have to then compare in a different part of the cosmos, how does life compare to life on earth, but how does it also differ? An abstract algebra is basically like that. We were basically astronomers who also are algebraist. We know our native algebra, but we're trying to then experiment and learn about extraterrestrial algebraic structures. And some aspects to be very different, they'll be very much the same. But in some aspects, they'll also be very, very alien. An abstract algebra is all about studying both the similarities and the alien properties of these different algebraic studies. And it really is a sci-fi version of mathematics, but like any good sci-fi, it's going to be a pretty fun experience. And so I'm glad to have you along here. That brings us to the end of our lecture, lecture number six, that also brings to the end of chapter two in Judson's textbook, speaking of aliens, that we're going to have our first encounter of the first kind, you know, next time we start in chapter three groups. So stay tuned for that one.