 Let the fun begin here, okay? So I'm going to We don't have a whole lot left to do but we do have to shut it Okay. Thank you Okay We're gonna finish up 3.2. It's gonna take very little time really and then We'll talk about the exam. I'll pass your homework back at the end of class today Homework at scores actually pretty good. I'm not sure if that's because More of you are leeching off of people that know what they're doing or if you're putting forth a little more effort Are getting getting the hang of this stuff? Hopefully it's a ladder of those But yeah overall the scores were pretty good. I was pretty happy with your work on this homework. So I Said I was pretty happy with your work on this homework Overall so very good All right, so let's let's just finish this up and then we will Talk about the exam And remember of course you should you guys are all here. So you know this course There's a exam on Thursday. So make sure that you're here on I know Make sure that you're here on time So that you have The full period to get this done Okay So let's see where we're at here So we're just going to finish this up. Okay, so we were proving a proposition Okay, so it's been a few days here. So let's just Remind ourselves what this was and the proposition was that the nth prime piece of n is Less than or equal to 2 to the power 2 to the n minus 1 Okay, so I'm not going to go through all of this again So let's just recall where we left off. Hopefully I remembered this so I tried to convince you that P sub n plus 1 Was less than or equal to The product of the first n primes and then plus 1 Okay, I think I stated that I don't know that I got too much further than this, but you guys have this in your notes Yeah, okay All right, so this won't take too long. We'll be done here pretty soon So p sub n plus 1 is less than or equal to the product the first n primes plus 1 and Remember we're using strong induction. I'm not going to write this down again the strong induction So we're assuming that this claim holds for p 1 p 2 p 3 all the way down to p yet, right? Okay, so let me just write this down just to be clear so We know that p sub 1 is less than or equal to 2 To the 2 to the 1 minus 1 right p sub 2 It's less than or equal to 2 to the power 2 to the 2 minus 1 p sub 3 is less than or equal to 2 to the 2 to the 3 minus 1 and Then of course keep going on down to p sub n is less than or equal to 2 To the 2 to the n minus 1 Okay, so let me put I'll just put a little little box here to offset this Okay, so from these two facts we get Okay, so p sub n plus 1 is Less than or equal to So notice we have we have this each one of these primes is less than or equal to one of these corresponding powers of 2 Okay, so what we can say then is that p sub n plus 1 then is less than or equal to This times this times this all the way down to this plus 1 right because each of these primes are less than or equal to the Corresponding power of 2 on the other side, so we can certainly do that, right? And so let me just write this out in a slightly better way So what is the first one simplified to well 1 minus 1 is 0 2 to the 0 is 1 so this just becomes 2, right? So we have p sub 1 is less than or equal to 2 to the first All right, what's p sub 2 less than or equal to all this is just going to be 2 Squared all right, and the next one is going to be what is this power of 2 for 3, right? So this is all right it this way though, so it's 2 Squared right You guys see what I'm doing here This is just 2 squared each of these primes are less than or equal to the these corresponding powers of 2 And then we're going to just keep on going down the last one being 2 to the 2 to the n minus 1 and Then of course don't forget right. We still have this plus 1 at the end, so I'm I still need that plus 1 Okay Well, what do we do when we have the same base and we multiply all of these guys together What do we do with the exponents because I can rewrite this part as just 2 to a single power, right? You add you add the exponents you guys know that now, please so this is It's going to look a little nasty, but 1 plus 2 plus 2 squared plus on down to 2 to the n minus 1 and Then of course again, we saw the plus 1 on the end, okay? And so I'm going to sort of offset this just so you can see where I'm going now. I'm just going to put this in parentheses 1 plus 2 plus 2 squared Plus on down to 2 to the n minus 1 is actually just equal to 2 to the n minus 1 This is In case you're wondering This is from this is actually a problem that you did. This is number 2 from section 1.1 I'm not going to re prove this now But you actually did a problem that that dealt with coming up with nice closed formulas for the sums of powers if you did Those of you that took calc 2 probably with series Geometrics it may have been a long time for you But you may have seen this a while ago that you can condense these things down This is kind of how you you get you find the limits of these geometric series you do stuff like this, okay? So and if you if you forgot this that's okay. That's fine, but just Accept that this that's what this is equal to so this becomes 2 to the 2 to the n minus 1 plus 1 okay All I did is I just replaced this with what it's equal to here. I got that okay You buy this do you believe that It's a really long one. Sorry, but Well, what what am I saying really I'm just saying that 1 is less than or equal to 2 to the 2 to the n minus 1 That's all I'm saying. That's definitely true, right? What's the smallest this can be ends a positive integer, right? What's the smallest this can be 2? Right Okay, so Here here. This is what I mean. I'm not writing the parentheses like I was before That's what I mean. Okay, so the power of 2 both of these powers are 2 to the n minus 1 Okay, and the minus 1 is not with the n up here. It's it's outside Okay, well, what is 2 to the 2 to the n minus 1 plus 2 to the 2 to the n minus 1 it's 2 times 2 to the 2 to the n minus 1, right? There's two of them, right? We just multiply it by 2 What do we do here? You guys agree that this 2 is the same thing as 2 the first of course you should agree with that So if I multiply these together, what does this become? 2 to the 2 to the n right because you have the exponents and the 1 and the minus 1 will cancel out and you get 2 to the 2 to the n and Would you agree this is equal to 2 to the 2 to the n plus 1 Minus 1 right, okay This I didn't do anything. This is a 1. I'm sorry This isn't anything complicated n plus 1 minus 1 is n so we certainly get the same thing that we got over here Right. Why did I write it this way? Think about this is the last part of the proof. We're done now We're done. What is the inductive step? What do we actually have to show? I didn't write it down because of space here but if we're trying to prove that p sub n is less than or equal to 2 to the 2 to the n minus 1 and We assume it's true for everything up to n. Well, what is the inductive step? We have to prove that p sub n plus 1 is less than or equal to 2 to the 2 to the n plus 1 minus 1 And that's exactly what I have here. That's why I wrote it that way because that's exactly what we have to establish in the inductive step Okay, so now you can see explicitly that we we got exactly what we needed when we replace the n with n plus 1 So this takes care of it now So may have a question but what I'm what I'm saying though is is and that's what I have That's what I have right right here, but what I'm doing is I'm I'm I'm just carrying it farther so you can see explicitly that we actually have exactly what we need in the inductive step Okay, the reason why I did this is just to show you that we can actually I'm just I'm just caring to prove through so I You know cover every little detail here This is not I mean this is not technically really the end of the proof right because what you really have to show Is that piece of n plus 1 is less than or equal to 2 to the 2 to the n plus 1 minus 1? We don't quite have that yet But once I do this now we've got any other questions here. Yes Yeah, yeah, I mean so yeah, I mean you're basically wondering well, how do you go for how do you get from here to to this point? Yeah, okay, here's yeah, here's here's that's a good question the answer is you need to get I mean what we really want is we want to get piece of n plus 1 less than or equal to like I said 2 to the 2 to the n plus 1 minus 1 which is the same thing as 2 to the 2 to the n Right that that's where you I mean you know you need to get there for sure The inductive step is piece of n plus 1 less than or equal to 2 to the 2 to the n Right, this is what we get when we replace n with n plus 1 and so what we've got right now is this So what we need to do is make sure that this is less than or equal to 2 to the 2 to the n And the way to do that is just to notice. Oh, well if we multiply this by 2 then we'll get 2 to the 2 to the n So as long as this is less than or equal to that then that's exactly what I end up with is multiplying it by 2 So basically what you want to do is you want to write an inequality that Will allow you to end up with that at the very end and the way to do is just to notice that one is less than or equal to this And then you get it to pop out at the very end That's not a totally satisfying answer But that's that's really kind of sometimes you just have to sort of be a little bit clever Just kind of to mess with it a little bit until you kind of figure out what the right thing is Yes If you were if you were to do this yourself, then yeah absolutely absolutely for sure I mean a lot of proofs that are not so straightforward You're not gonna see exactly how to do it right away and you're gonna have to fiddle with it for a while And then once you see how the pieces fit together, then you write the finished product down. Yeah Yep Any other questions here? Are you gonna have to prove this on the test? No No, okay. Well, um Yeah, I mean I basically agree with you with what you're saying. I mean Yes, I don't know. Did you have a question or? Okay, no, I I I do not in general think that My style is not to put really really tricky problems like for example suppose I didn't do this I'm not gonna have you prove this on the test by the way suppose. I hadn't done this before and I asked you to prove this What a on an exam? Well, yeah, I think that would not be totally reasonable because there's a lot of tricks going into this I wouldn't expect you to come up with this on the test So no, I wouldn't put something like this on the exam. Okay, that doesn't mean that there aren't gonna be any proofs But now of course how hard something is and how tricky something is these are subjective things, but Things that I think you should know how to do that aren't gonna require a lot of work. Yeah I mean those of you those are certainly fair game Any other questions about this? No From the last test. Yes No, you should you should know how to do. Yeah, you should know how to do the problems The solutions are all are all posted for sure Okay, so I'm not gonna narrow it down any more than that just but I'm gonna tell you that yeah one of them will Will be directly from the test So no, so my point is learn what you did wrong And this is gonna help you for the final exam. It really is gonna help you on the final You're gonna see some problems on the final that you've seen before too. So this is gonna keep you You know up to speed on this stuff. Yes Yes Yes The reconciling is that you have a big Portion of your grade is homework. I think I'll be honest with you Here's here's part of the reason why the performance was so was as poor as it was a lot of you guys are getting help from from other Sources and are not actually understanding what you're doing. There's no doubt that that's happening There's no question So in some sense, and I know some of you're gonna get mad at me for saying this the scores shouldn't have been as Low as they were that is definitely a huge function of why they were as low as they were There's no doubt especially looking at the definitions. Most of you miss the definitions. That's bad I mean, I wrote them down explicitly Okay, and not necessarily missed all the points, but you missed a decent number of the points My point is you got to know how to do this stuff yourself. You really do you have to okay? I mean with the scores horrendous. No, they weren't horrendous. I mean most people passed the exam The scores were not as good on average as the homework That's and I think just seeing similarities between a lot of your homeworks I can see that some of you are working in groups together, which is fine But some of you are not quite catching on to what you're supposed to do on your own And then we go on the to the exam you have to do it on your own you're sort of tripping and so I mean My answer to your question is I'm not gonna change anything because I don't think I've done anything inappropriate here not at all In fact, I think I've been very generous overall, especially with the amount of weight I'm giving on the homework Okay, and the completion grades. I mean I'm being very nice to you guys I could I could hammer you so much more if I wanted to I'm just being honest I could and it would be reasonable. So you guys have got it actually pretty good. This is just a more difficult course Okay Yes Well, I so here's the thing. Okay. Yes, I have the problem is that There are basically Almost no people available to do the SI and the one person I know that's available I think is gonna screw you up more than is gonna help you. I'm not kidding you I'm not kidding you. I'm not gonna say who it is I'm I'm being serious. I'm being very serious and of course, this is no work for me I'm doing I'm so my suggestion is you know, if you're having trouble come and talk to me email me I told you in the beginning that I will I will work with you I'll set up appointment times to meet with you outside of my office hours I am happy to help you out. You just have to take the initiative and say hey I need some help. That's all you have to do And I will help you I'm definitely more qualified but then any of these SI folks are to help you I assure you of that Okay, you do not want someone who's never taken this course trying to explain proofs to you I guarantee you do not want that that's not going to help you. In fact, your score scores are gonna go down probably You do that. Okay. I know I got you got to trust me a little bit here. I know what I'm doing here I have thought about these things. Okay, I'm not out to screw you guys I'm not to help you but you have to take advantage of it. You guys just have to take advantage of it okay, and Some of you a lot of you are I get a lot of people come to my office and I'm giving you hints I'm you know helping you and everyone's welcome to come in, you know, I like you guys I want you guys to do well, but you got it. You got it, you know, put forth the effort. That's all I'm saying. Okay All right We can we can talk more about this here in a second. I just want to finish this up and then we can talk about the exam Okay, this will take not much time at all This tells you Something about the distribution of primes. It's not very good, but it's okay Okay, this is the last thing that we're gonna do Okay, so right clear on this the power of two is two to the end. That's the power on two. Okay Okay, so This is all there is to it Let M be a natural number. It's basically one line P1 is less than P2 Which is less than P3 On down Which is less than P sub n plus one Okay, this This doesn't require any proof. This is just comes from the ordering of the primes right P1 is the first prime It's two P2 is the second prime. It's three P3 is five So there's there's nothing to really say here. This just comes from just what P sub n means, right? and What does the theorem say P sub n plus one is we just prove this less than or equal to two That's a big two to the Two to the end Okay, I could if you wrote the notes down Just see what what I did at the very end right piece of n plus one was less than or equal to we went all the way down The very very end was two to the two to the n plus one minus one Right, that was the last thing you should have in your notes before this corollary But what's n plus one minus one it's in right? So this just comes right from the right from the end of proposition two Okay, so Let me just write this down Okay, so now Notice though that this doesn't quite prove that the corollary right it the corollary says that there are least n plus one primes less than two To the two to the end well, I have n plus one primes here But the last one I have less than or equal to I need to rule out the possibility that it's actually equal Right because if it were equal then I'd only have n primes from this list that were strictly less than two to the two to the end So I need to know that that piece of n plus one is actually less than it not not equal to two to the two The Mersenne primes Oh, it looks it. Yeah, it looks sort of similar to this Yeah, that's something that We'll probably get into later. It's not it doesn't Yeah, I mean so the point is that there's you can actually find out some primes that are powers of two Plus or minus one Two squared minus one to cube minus one not two to the fourth Two to the fifth minus one. So it's kind of this observation that oh wow powers of two plus or minus one seem to generate a lot of primes So so yeah, I mean it's I don't know if it if it comes directly from this corolla I don't really know what the history is exactly, but I'm guessing they're not really related But this is something we'll probably get into later Right, right two to the fifth minus one give you the prime Yeah, okay, so piece of n plus one is actually not equal to to The two to the end. Why is that? Why is that? It's not a really it's not a very difficult reason Think about what piece of n plus one is Yes, that's all it is Okay, remember these guys are all primes now piece of n plus a prime number two to the two the end is definitely not prime Right no matter what n is okay, and it's positive, right? It's a positive integer. So so two to the two to the end is Has two as a factor, but it's also bigger than two. It's at least four, right? So it can't be can't be prime Okay, something I'm gonna say let me hold off on this Let me just finish this first since piece of n plus one is prime Two to the two to the end isn't isn't prime Okay, and That's all I'm gonna say really I could say one final sense to sort of tie all this together But the point is now since I know that piece of n plus one is actually less than two to the two to the end now I've got n plus one primes that are all less than two to the two to the end, which is exactly what we wanted to prove Right Does that make sense? Okay So that's that's it So what I want to do do things I want to do with the rest of the time I want to talk about things to study for the test and then I want to give you some some homework hints and maybe go back over a Couple problems. I didn't grade just to kind of give you an idea of how to go about doing those the solutions aren't posted online Okay, so Again as I said The exam is going to start with section 2.4. There's not a whole lot of stuff on this that's going to be covered on this test Let me reiterate one more time one of your One question from the last exam will be on this test So you should know how to do the problems on the last exam That's something that you guys really ideally should be keeping up with anyways when you get your exam back You should be going over it and looking at the things that you missed Some of these will be on there on the final some of the questions on final be taken directly off things that you've already done Okay, so even more so probably than them in the exam. So this gives you motivation to be prepared for the final Okay, so 2.4. I'm just going to sort of say this. This is the Euclidean algorithm. I didn't actually do the Euclidean algorithm Basically what we did the new stuff in in section 2.4 that we did was the least common multiple, right? You should definitely know at least common multiple is if I could ask you that on the exam What what is given to non-zero integers a and b what to find the least common multiple? What does that mean, right? You mean what do you mean? Do you mean actually find the least common multiple? I mean, that's that's possible, but I mean I could say something like here are two integers What's the least common multiple? It's not gonna be anything nasty like 1,050 and 2,145 or anything like that Sure, it's possible that I could say something it would be very basic It would be something of this kind that you would have seen in sixth grade or something like that, you know So What else And that's that's the main thing that you that you need to know As far as theorems are concerned You should know and I'm just so I can save time in here I'm just gonna say this and you can write it down. I assume you all have your book You actually own the books right or at least you have a copy You should know theorem 2.8 on page 30 theorem 2.8 on page 30 again I don't there's no name for this, but I might say you know something like You know Prove some basic property about the GCD or the LCM you're free to use these things so the reason why I'm saying you should know this I'm not gonna say what is theorem 2.8 on page 30 that will not be I won't say that on the test But if you know it it might help you with a proof for example, okay So theorems that I've done in class. You should definitely be familiar with is what I'm getting at here. Okay, and The corollary also on page 30 you should know that too. So there's just a theorem in a corollary. There's no ambiguity there It's just that's that's all there is on page 30. So you should know those things Okay And of course and this goes without saying you should be familiar with how to how to work with these concepts You know, I could I could ask you to do a you know a problem or a proof That's relatively straightforward You should be able to do those so you should be comfortable with the homework as well, right? Okay, we skipped 2.5 3.1. Okay, things you should know in 3 1 You should know the definition of a prime number and I mean really the definition Rigorously there were some issues, you know, there were some issues on the first exam with definitions. So What do what do we need for an integer I may say, you know, for example, I could say what is it? You know, let let M be an integer to find what it means for M to be prime Well, there are several things you need you if you just say the only devices are one in itself. That's not right Because it could be one. You only do the only sorry the only positive devices are one in itself, right? Well one is not prime the only positive device or one is trivially one in itself, right? But it's not prime. So you need to make sure that you get all of these little details, right? An integer M is prime if and only if M is bigger than one and the only positive divisors of M are one in M That's exactly what it means to be prime So don't leave out those details some of your missing points because you're leaving these little details out You can get this guys. You guys can all do this. I am confident you can do it Just make sure that you know it. There's not a lot of there aren't a lot of definitions here I would expect and I will ask you definitions on the test. I would expect that you all get them really There's not a lot going on here Composite well Okay That's pretty easy. That's the definition of composite is that M is bigger than one and it's not prime. That's the definition That's all it is not prime There was a lemma I proved about composite numbers if I don't if you recall this an integer M bigger than one is Compositive and only if M equals R times S Where R and S are strictly between one and M our integer strictly between one and M? I'm not gonna ask you that I'm gonna say what was limit to on you know February 27th I'm not gonna do that. Okay, but you might want to be familiar with it again just because it might What's that? No, that limba was not in the book It's just something that I should I proved to enable us to prove the fundamental theorem of arithmetic So no that wasn't in the book probably that's not gonna come up on the exam But anything that I've proven, you know, especially theorems. These are these are things that are certainly fair game. You should know Okay, um So you should be familiar with Theorem 3.1 and the two corollaries on page 40 theorem 3.1 and the two corollaries on page 40 This is of course this stuff I've done in the notes as well on page 40 Okay, also you should definitely know the statement of the fundamental theorem of arithmetic. That was the main theorem of 3.1 I will definitely not ask you to prove it. Okay, you do not have to prove this That could take the entire time to do and I'm not gonna do that. I don't want to grade it either so But you should know what it says for sure and it's not It's not that bad now now You can say it a couple of different ways I phrased it slightly differently than the book did and see you might say oh, well, which one should I know yours or the book? I don't either one's fine. It doesn't matter. Okay So the point is the fundamental theorem of arithmetic says that Every integer bigger than one can be written uniquely as a product of primes and the unique part just means up to the order of Factors right two times three and three times two of course are the same give you the same thing Another way of saying it and this is what I did in class Which I think is a slightly more succinct way of putting it every integer and bigger than one can be Written uniquely as a product of primes in canonical form Okay, then there's no ambiguity about what do I mean by order and all this stuff? Okay, so Should you? Well, let me say it this way. You should know what canonical form is too I say, you know, what does it mean for for an integer bigger than one be written as a product of primes in canonical form? Well, it means that You are collecting all the primes together. This is in your notes, too, right where the primes increase in order basically That's the idea right p1 to the alpha 1 times p2 to the alpha 2 times p3 to the alpha 3 where p1 is less than p2 Less than p3 etc on down Is that clear? Is that you guys okay with that? You know what I'm saying? Okay? Okay And then 3.2 Okay, so This you know this this sieve problem I May ask you okay, here's something that I may do okay I'm not going to ask you to find the sieve of Eritastans because it's just a little bit bulky and awkward and I'm not going to do that What I may do though is say something like this I may give you and don't worry. This is not going to be anything huge. Okay, it's not going to be that big In fact, it'll probably be relatively small in fact But I may give you a number and say okay Is this number prime and It's again, it's not going to be anything big you probably only have to check maybe four numbers or something like that But what what what am I getting at here? Oh? No, you don't have to do that. I mean So you could well you could you could do that or you could use remember There's there's a proposition I proved that said that if you have a composite number n Then it has a prime factor less than or equal to the square root of n right So if you want to see if the numbers prime you just have to check the prime factors less than or equal to the square root of That number and if none of them actually are factors, then you know it's prime I may give you a composite number though, so you know I may not give you anything But this is something that you should be aware of So this sieve right you could do that you could you know there are various things that you can do And if I give you a computational problem like that I'm not going to get all picky with your work because you're not writing a proof You're just really trying to compute something and see if it's true or not okay, so I'm not going to get picky and say Oh, you didn't write this out rigorously or anything like that You know when I say a small number I mean literally it's not it's not going to be something like five thousand or something like that I mean I may give you you know I mean it's not going to be this but you know I may give you you know 113 or something like that right so it shouldn't take you very long to go through Right, so I mean you know Without a calculator, you know the square root of 113 is somewhere between 10 and 11. All right tense You should know that I mean that should be something you can do without a calculator So now you only have to check a few primes and then you're done Okay, well No, but I mean it's it's going to be something that you really should be able to do without a calculator So it's not going to be big big numbers. Okay. All right And then you should know you know the theorem that I that I prove for you So I you know I could ask something like in the corollary. I could say okay Well, how you know there are at least how many primes less than or equal to 2 to the 2 to the n for example n plus 1 So you should be familiar with that I could ask you something like that or I could say you know Are there infinitely many primes are they finally many primes? So I'll tell you this is one thing. I'm not going to make you write down any proofs of any theorems from the text. Okay What I want you to do is know what they're saying All right Yeah, you should definitely know that's you already know this you already know right now if I said are there infinitely many primes or finally Primes all of you know the answer to this You don't need to study this, right? Okay So and then aside from that Again, you should be you know You should be familiar with You know doing some some proofs because you're definitely going to have some proofs That's all of course. That's all point of this course is writing proofs. So you're gonna have to do a few of those Okay So what I want to do now and then of course you're welcome to ask some questions too as we go through this I would like to talk about a few problems, especially a couple from the homework That's due on Thursday that you might have questions about Okay So let's start With the most recent section. Let's start with section 3.2. Okay, so you had to do number 3 and Number 3 says suppose that Let me let me make sure that I ran it down that way and so this just says that given That p does not divide n for all primes p Less than or equal to the cubed root of n Show that n is the product of two primes. Thank you for sharing that Yeah, actually Actually, yes, because I would have done it really quickly. I would just go like that and it would be over Okay, the hint says assume to the contrary then contains at least three prime packages this hint is not that good I Would actually advise you to ignore the hint and I'm gonna I'm gonna Do something I think it's a little bit clearer than this If you say well, okay, if it's not prime or the product of two primes that does not mean That does not mean that it's a product of exactly three primes Right, it just means it's a product of more than two Maybe it's 17 primes and then you just get all convoluted and confused very easily if you start worrying about that stuff So really I think the hint should have said this the hint should have said assume I'm gonna write this down but assume that n can be written as the product of three integers bigger than one That's where you're gonna get the contradiction. Okay, you can't write it as a product of three integers bigger than one And then from once you know that then it automatically has to be prime or the product of two primes Because the product of more than it's then it's okay Let me just do this first. I'm getting ahead of myself Okay, so here's the idea, okay I'm not gonna write out all the details. I'm going to leave this to you to write out some of the details Suppose and bigger than one is an integer you should start off the proof like this and And P does not divide n for all primes P less than or equal to the cubed root of n This is sort of the analog though of it. So if you remember this this I proved this for you actually in class that every composite number Has a prime factor less than or equal to the square root of itself, right? This is kind of the analog of that except now we're dealing with cubed root instead of square root Here is the claim and this is what I really Suggest that you prove that that you prove this Sometimes I'll say in class. Okay, here's one way you can go with the proof. It doesn't mean it's the only way Okay, it's in some of you of course have not done that you've done you've done the proof a different way And that's fine. Okay, a lot of times there are three or four natural proofs of a proposition So you don't have to in general do what I'm suggesting as long as it's correct This I think is the shortest way you can do this problem. I really think it is so the claim is that P is not The see if I can finish this down here not the product of three integers sorry, that's not what I meant and It's not the product of three integers Greater than one What I mean is that all all three integers are greater than one. That's what I mean when I say that Okay, so what I'm what I'm saying one good a nice strategy to prove to start the proof is to show that okay So the assumption is that P does not divide in for all primes P less than or equal to the cube root of n What I'm claiming is that that implies that n cannot be written as a product of three integers all of which are bigger than one So I think you should start by proving that that n cannot be written as a product of three integers that are all greater than one and I'll give you an idea as to how you go about this. Okay, can I go on to the next page here? Okay, so if you want to prove this claim Suppose the way of contradiction We can do this and can be written as a one times a two times a three Where a one a two a three are natural numbers and They're all bigger than one. Okay, so we can certainly assume That they're in increasing order right a one's less than or equal to a two and a two is less than or equal to a three right Anytime because multiplication is commutative. You can always just arrange them from smallest to biggest like this Here's all I'm going to tell you and I'm going to let you finish this yourself note a one times a one times a one is less than or equal to a one times a Two times a three think about this for a second This isn't all that complicated. You guys see how why this is true a one's the smallest of them Right, so a one times a one times a one certainly less than or equal to a one times a two times a three because This is bigger than equal to that. This is bigger than equal to that and the first ones are the same Yeah, yeah, okay, so I mean here's Okay, so you Really kind of want to go about it like this and I'll tell you the reason why Because if you do your proof then all you've really shown is that and that end can't be written as a product of three primes But maybe it can be written as a product of nine primes You haven't completely precluded that all you've done is shown that it can't be written as a product of three primes So I'm not going to assume that they're prime. I'll but that's a good question. I'm going to say more about that in just a second, okay? okay, so if you go back to your notes if you recall the proof that I did that if And if n is composite then n has a prime factor less than or equal to the square root of n if you go back to that proof It's now at this point. It finishes off very similarly to that. In fact, it's really the same thing It's just now you have three you have a cube instead of a square This is in your notes. You may not know what I'm talking if you look in your notes You should be able to finish this proof. You just mimic exactly what I did at this point Now what I'm going to say is well and this I'm just going to say this is up to you to write this down But I would encourage you to pay attention to this Well, why does that prove everything? Why does why does then n have to be prime or the product of two primes? Why is that? Okay, and I really want you guys to listen to this, okay? So what after you finish this you will you will have shown that n cannot be written as a product of three integers That are all bigger than one, okay? So what you're trying to prove in the end though is that n is prime or the product of two primes Well, what if it was a product of more than two primes? Hang on a second if it's a product of more than two primes You can certainly group it into the product of three integers bigger than one You see that suppose it's five primes We'll just take the first one the second one and then all the product of the rest is your third one You see what I'm saying. That's why this is the way to go. Okay Does that make sense? So now this sort of catches it now it catches everything now now You don't have to say well more than two could be three or five or fifty or ninety This takes care of all of it right away because you're not assuming that these are all primes now You can get it for free at the very end Okay, so the point is this is the end of that what I'm going to say If you know a number can be written as a product of at least three primes Then it can certainly be written as a product of three integers that are all bigger than one Okay, and you know you can't do that. So therefore it has to be prime or the product of two primes That's how you finish the proof Okay, okay, and again, I'm not saying this is the only way to do it But this is just one way to do it. That's pretty short and pretty succinct Okay, any questions about this problem? So I think for a this is about the square of p being irrational Some of you maybe did this in discreet before I didn't talk about about this I'm gonna I'm gonna say something about about this now Okay, so first thing I'm gonna tell you you've you've if you took discreet you all got this definition You all knew this before this course. I'm sure but let R be a real number So now we're going outside of the integers for a second and R is rational if So what's the definition of rational me tell me how to finish this Joe, right? Yes, if R equals p over q for some Right, right some integer p and q Not zero right, okay. I assume most of you have seen this before probably in discreet or before that maybe Okay So irrational I'm not gonna write this down, but of course a real number is irrational if it's not rational right And so the point is that the square root of p is a prime the square of p is not rational That's what you're trying to prove here. Okay, there's really here's the hint The book kind of goes through and proves the square root of 2 is a rational Of course, you can look up this stuff online and this is very common popular well-known stuff Okay, but here's what I expect you this this if you do the proof this way I would expect that you would write this down in your argument somewhere. Okay, so the hint I'm gonna give you is Basically this hint every rational number Can be written in lowest terms you should Prove this Okay, does everybody know what I mean by this rational number lowest terms means that You can write it as a fraction every rational number can be expressed in a fraction where the GCD of the numerator and denominator is one That's what I mean, right Okay, can you and you can prove this very easily using what we've already done in this course about with the GCD Okay, and I'll give you a hint here Okay, I don't want you to make this really kind of informal and ad hoc and say just factor out factor out the stuff And you can do it and cancel it you can make this very rigorous in about one line using things We've done in this course and here's here's what you're gonna. Here's what you're gonna use. Okay Let D be The GCD of we'll say P and Q Again, I'm I'm not being completely formal now. I'm just giving you that the ideas of the pieces here Okay, think about this Tell me if you buy this P over Q Is equal to P over D divided by Q over D you guys buy that Multiply through by D you get P over Q right and There's a theorem that we talked about in class and maybe you've forgotten this and that's okay if you did but if D is the GCD We're using P and Q instead of A and B But if D is the GCD of P and Q then the GCD of P over D and Q over D is 1 if you divide through by the GCD The GCD is 1 okay, so now you know that This rational number can be expressed as a fraction where these the numerator and denominator are relatively prime Okay So you see you can do it now just quoting what we've already done in one line and then So Let me just say this first I'm just I'm just telling you to two facts I'm just saying that one that whatever our given fraction is can be certainly be written in this way They're the same thing and the second is that in fact these guys are relatively prime Which is exactly what it is that we're trying to establish is that every rational number can be written as a fraction where the numerator Denominator relatively prime. Yes Oh, no, no as long as you're as long as your proof There's definitely more than one way to do this as long as your proof is correct and just uses Things that we've done in class. It doesn't go outside of what the scope of what we've done then it's fine So no, I'm not saying that you have to do it this way. I'm just saying that this is one way you can do it Okay, and then once you've got this if you guys seen this the proof the square to two is a rational You've had to have seen that in discrete math at some point Yeah, so it's the same idea here okay one way you can proceed here And this is I'm not going to say anything else other than this is to say by contradiction assume the square to p is rational and So then by this you can say the square to p is a over b or p over q where a and b are relatively prime If you square both sides you'll and you use some facts from class you can get a contradiction Your contradiction will be that in fact a and b are not relatively prime your contradiction will be that p divides a and p divides b That's the contradiction Then you're done Okay It's very similar to one of the standard proofs that the square to two is irrational, but again There's more than one proof of this so you can I'm not saying you have to do it this way, but that's one way to do it Yeah, okay, so the here's the way you proceed with here's one way to proceed with the proof just one way It's to say okay suppose the square to p was rational It's not but suppose it was then you'd have the square I'm just gonna jot this down but down here just okay if it were rational Then you can say the square to p is equal to say a over b where a and b are relatively prime right Okay, that's how you would start Then what can you do? Okay? I'll just I'll just say this if you square both sides you get this, right? I expect your proof to not look like this. I'm just giving you the details, okay? And then if you shuffle all the algebra over then you get p b squared equals a squared, right? I'm not going to go any further than this, but now if you there's one theorem that actually you we did in 3.1 Section 3.1 that will allow you to conclude something. Well, you know p divides a squared now, right? Then you can extract even more information from that because p is prime That's all I'm gonna say and then then it's Yes Well, that's ultimately where you're gonna want to go. Yes Well, you have to do something a little bit more than this, but I'm I'm I'm gonna stop here because I've already given you If I say anything else, I'm gonna give the whole thing away. I don't want to do that. Okay, so Think about it. Think about it Okay, you got you got to do this. I mean you got to work on it I'm not gonna I'm not gonna finish the proof if I if I answer anything else I'm gonna give the whole thing away, and I don't want to do that okay all right, so I Also wanted to talk about Something that I'm gonna pass your homework back here in a little bit, but I'm gonna say something about I'm gonna ask you guys a question here. Do you guys have this written down now? What about three? Oh Okay, no, it's actually three is involved and that's weird actually three does come up in this question Okay, so Here's my question suppose that X is an integer and Three divides x Natural natural number. Yeah, sorry natural number and three divides x is Just this is all you know. Okay, so the question is do you is x composite? No If x is three it is not composite. Okay a Lot of you and I'm gonna pass the homework back in a minute. There was this problem Where Let's see basically you you I think it was piece of prime bigger than or equal to five So the p squared plus two is composite I think that was the problem a lot of you just said all three divides it so it's composite No, that's not necessarily the case Okay Be really careful about this just because a prime number divide something does not make that thing composite because it could be the Prime itself Okay, in order to know that a number is composite three to you know three dividing x doesn't necessarily make x composite It does if x is bigger than three then it makes x composite because then it has a factor other than one in itself Right three sandwich in between one and x. That's what makes it composite Okay, but just because something bigger than one divides a number does not make it composite by itself You need to say something else you need to say that it's actually bigger than that number for example, then you get composite I Saw a lot of this on the homework a lot of this Well, and I think you guys know this but again you're writing proof So you need to make this rigorous three dividing x does not imply x is composite counter example x equals three right Okay, you might say I'm being overly pedantic here, but That that's the whole point of writing a proof is that you you you you Establish everything completely rigorously So some of you are going to see that you have some points taken off on this problem for that reason Okay, if you noticed Okay, so this this I'm just going to say I'm not going to write the problem number down But the problem was p is bigger than or equal to five show that P squared plus two is composite Okay, now there was a hint That said that you know hint p has to have the form 6k plus one or 6k plus five if you remember this This is the hint. I remember I said this a lot before so you guys have to know that this is fair I told you if you'll remember this now, right? Have to prove hints. I've always told you that some of you didn't do that you lost a few points No, no, I've said you always have to prove hints you guys you guys know I've said that I've always been saying that You can use previous stuff, but you have to prove the hints now Yes, yes, yes, exactly. Okay. Well, all right Yeah, right If I if I have not told you that you don't have to do it then you do have to do it, okay? Well, okay, I'm not going to write all this again out in detail, but but this is important, okay? Because of time I'm I don't have time to write all the k's and integer blah blah blah blah blah, but okay some of you said well P You sort of crossed these out right away Because you say well these are not these aren't primes because well for example here We can factor out a two we can factor out a three we can factor out a two But the point the but you never use the fact most of you in fact Maybe all of them are not all of you some of you do this you didn't ever use the fact that p is bigger than or equal to five That does play a role in this somewhere, right? Okay Because for example two two squared plus two is sorry So Three squared plus two is not composite Right, so somehow to do this proof correctly You have to use that fact in your proof somewhere that p is bigger than or equal to five most you did not it never came up Because some of you might say well this this is you can factor out a three so this this is never prime No, if k is zero it is prime You see that some of you are missing this, okay? If k is zero this is prime So a lot of you are missing that those steps in the proof Okay, so be very very careful when you're doing this to think about it and say ah Just because you can pull something out does not mean that that that thing is not prime does not mean that Okay, so I wanted to address this Let's see I have time for anybody have any any questions they want to ask a gun No, I don't I Do not have a gun Who's the gun for okay, I Okay, I do have to say that especially give me your comment guys. This is a proof class You require some thought I'm not a big bad meanie. So you think that I am I'm not this class is hard. It's a hard class. Okay, I Hope you didn't say that you wanted to gun to shoot me because I'm actually one of the better instructors in this course I assure you of that You don't think it you're wrong. I'm sorry. It's true. Yes Yes Yes, that's yes Right because yeah because you can pull out of three Since three divides so P is of this form you can pull out of three three divides P and P is bigger than equal to five Then P is it then three is a factor. That's not one and not P So therefore it wouldn't be prime and that's how you rule it out Okay So okay any other real questions here. I wish I did I Don't know Of course if I tell you it doesn't matter. It's not gonna affect your your studying strategy, right? I mean having having the knowledge is I mean you there will be proofs on the exam It'll probably comparable to what we did before I'm not gonna say I see Okay, yeah Well, so well, I'm not I'm not gonna say I'm not gonna say Yes From from before you mean Okay, I know I mean everything that I would ask you should come from what I told you before okay Yeah, yes But if I told you everything that's not gonna be on the test then I would have Then you could infer exactly what it's gonna be on the test. I'm not gonna do that. I'm a little I'm not dumb, okay? I Know I know that I know that I know I'm kidding. I'm kidding You I know you're trying I appreciate that I know what you're saying you're saying is there anything I can just say specifically that's not gonna be on the exam Well, yes, I mean you're not gonna have to write proofs of theorems from the text for example, that's not gonna be on the exam You know other than that You know that the statements of theorems the definitions and then you know homework problems that are you know not among the most difficult ones that the proofs are not extraordinarily long these are things that You know Maybe but possibly not okay But it should be it should be something that works out fairly quickly from things that you've done before and if you go back to the Solutions of the first exam that's more or less the case I mean even the induction problem kind of worked itself out as long as you know how to do induction You know and the other proofs were really pretty short, so it's gonna be stuff like like that Um Induction well, I mean so again there was an induction problem on the first exam This test aside, let me put it this way aside from possibly the problem the induction problem on the first test You shouldn't see induction on this exam Okay, so know how to do that first induction problem. Okay. I've I've I've decided yeah, I'll give you that Okay Well, let me go ahead and pass your homework back then Don't forget your next assignment is do Don't forget to bring it with you right on Thursday. Okay 3.2 3.2. Yes Okay, okay. Well, yeah, just send it to me as soon as you know, I just got it. Oh, I got you. Okay. That's that's fine Okay, okay, that's fine. I mean I it's just basically it's just up to you to keep to remember to get it done, you know Okay, let me turn this off here