Share on social media:

Motivate the Math Nostr Community
https://satellite.earth/n/MotivateTheMath/npub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99g

Wolfram Alpha
https://www.wolframalpha.com/

Sieve of Eratosthenes

https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

In this milestone 10th episode, we dive deep into the world of mathematics, exploring the fascinating concept of math as a liberal art and its implications in fields like cryptography. We discuss the importance of understanding math beyond its technical aspects, emphasizing the value of a strong liberal arts background for math students. The conversation touches on the societal impact of math education and how it can influence career paths, particularly in the realm of cryptography and Bitcoin. We also reflect on personal experiences with educational systems and the role of parental engagement in shaping curriculum and clubs, highlighting the potential for innovative math clubs in schools.

Additionally, we engage with complex mathematical concepts through listener boosts, tackling topics like prime power factorization, isomorphisms, and the sieve of Eratosthenes. These boosts challenge us to decode mathematical notations and explore the deeper meanings behind these concepts. The episode underscores the importance of demystifying math, encouraging listeners to embrace the language of mathematics and see it as an accessible and integral part of understanding the world. Through this journey, we aim to make math less intimidating and more relatable, fostering a community of learners eager to explore the depths of mathematical knowledge.

[00:00:04] Unknown:

Topic just in case we get to it?  

[00:00:08] Unknown:

No, man. The only topic is   is these boosts, I think.  

[00:00:12] Unknown:

Yeah. So, you know, look.   This is episode, I believe, episode 10.   Correct. About that? Okay. Wow. So that's big milestone. By my accounting. By my accounting. Big big milestones. So let's just acknowledge, man. Pretty pretty great.   I didn't know we I don't know if we'd do that. I didn't know if we'd get to 10 episodes. So this is great. Right?   This is, like, every fucking week, we print a block by doing an episode.   And   Interesting.   Like, we're starting to get   you know, we have the old school. Like, I'll say, like,   Dan, our boy Dan is, like, the old school,   student, the first student.  

Shout out to Dan, though. I saw him in person. I gave him a hug. Nice. Nice. He's great to meet in person always. That's how I know him really from, meetups.   But he's, like, first real kinda legit student   we that we   brought along here, right, who decided basically to dive into the rabbit hole a little deeper as a result of being motivated by Motivate the Math.  

[00:01:12] Unknown:

Well, in speaking of student, you you mentioned, other topics and stuff, and I will tangent here briefly is,   I just talked to   essentially where my kids are gonna go to school next year, and they are open to parental engagement.   Open to the point of   yeah. Just like the it's a smaller school, and so they're just like, hey. If you got something that, like, you're passionate about, you wanna do, I provided them a copy of my first Bitcoin, the diploma thing to be like, hey. Check this out. To that kind of parental engagement, like, curriculum engagement. Curriculum,   clubs, things like that. And I'm like, I this is I gotta start a math club there, obviously.  



[00:01:47] Unknown:

Well, hell yeah. So it's it's interesting. So I sent my kids to Waldorf schools. They require parental engagement, but they don't they're not open to curricular   suggestions.  

[00:01:56] Unknown:

Oh, no. Okay. Interesting.  

[00:01:59] Unknown:

I've had a you know, I've had an ongoing saga with now they already do really they, you know, they do a great job with math. They don't know it. They don't know it because they don't know that math is a liberal art.   You see this? Well   They don't understand that, and so they don't know how good they are with math.   And these are good. Because they don't understand that math is a liberal. They're great at liberal arts, and they're   you know, I've brought them through over the years. Just little things they do, knitting,   and things there's just things along the way that they don't realize. And my daughter who is, you know, math major will be the first to tell you, like,   it's at Waldorf school. If she didn't go to Waldorf school, she doesn't think she'd be a math major.  



[00:02:40] Unknown:

And there's this guy. It's my neighbor's son. I I I roll with him at jujitsu sometimes. He I was talking with that, and I was like, hey, man. Like, I had this, like I don't wanna say epiphany, but, like, I had this realization that, like, math is a liberal art. And he's like, yeah. I went to a liberal arts university and studied mathematics. And he does, like, cryptography and stuff. Right? Yeah. So there's other people that recognize this, which was which was cool to sort of, like, validate the thesis.   Not that it validates it from this random guy. But just hearing   this kid, he's like, oh, yeah. Like, I went to a liberal arts school. And he named that school, and I don't remember what it is because it didn't matter to me. I'm gonna say something kinda  

[00:03:14] Unknown:

controversial   maybe. But, like so, like, I think, like,   if you learned math as liberal art   and you end up in   so and you end up work let's say you just studied math, you end up working in cryptography. I think that there's a there's a I think typically   you will end up working for sociopaths   like chain analysis   or name your fork or, right, like Neil Kablex. Right? Just name, you know, name your fork because first of all, it's where like the money is and also, you know, that the money is not   in the,   in the liberal art side.   However,   there's another path. And if you really are if you are deeply grounded in liberal arts, there's a chance   that you may end up with us. There's a chance you may end up on this side   of   the math team   where we were just trying to   uphold   we're just trying to uphold,   I think, what Bitcoin promise is promising.  

Right? Yeah. And there's a part of it that has to do with   the society   not getting duped by   these by these people. These p these math students who are now being paid to weaponize their skills, then Yeah. You know, I think we're the defense. We're building the defense in that part of the Citadel.  

[00:04:32] Unknown:

Well and get this. So, like, I've I've had this long running thesis too. So geographically and it's I'm I'm not shy about it or anything. It's like ShenandoahBitcoin.club   is, like, my meetup site. Right? Yeah. Dude. Chad, that's Shenandoah, man. That's That Fuck yeah. But I've had this long running thesis too that,   you know, the people in this area, because of our proximity to DC and, like, just the the metro and all the three letter agencies and the bureaucracy and everything like that,   but the people that live out here in this valley,   they might commute over to that other side, but they I think part a lot of them get that, you know, there's there's, like, this swamp stench over there, and they don't wanna live there. They work there. They they mine their feet out there and everything like that. But I think   I think there is something to that. People the people that realize,   once they realize   that there's an alternative,   other than, like, paying into this giant, you know, war machine, this murder machine,   there there's better options.  

That's that's when they'll start showing up more.  

[00:05:32] Unknown:

For sure. I I don't like, I think if you learn math   in a in a way that's vastly not a liberal art, right,   I think it's just so likely you're gonna just submit to whoever pays you the   most, whoever, you know, just whoever can give   you your perception of the life you need.   If you and I think most people will do that regardless.   But the question is, is there, you know, when you're in it for five years, are you gonna be look you know, like, is it gonna kill you or are you gonna be totally fine with it? You see what I mean? And I think you have to have a strong if you have forget learning math as a liberal art. I just believe that people need a strong liberal arts and humanities   background.  

I just have thought a lot about it for my own children as, as math students. I think that's really important for math students, but it's important for everybody.   Right? Everybody I think who ends up in a job   with with that's,   soulless, should, it should kill them.   Right? And it should kill them fast.   It should start making them look for options fast.   And that's the best I think maybe we can hope for. Right? Is that, you know, we're we have built something by the time these guys have hit their breaking point.  

[00:06:52] Unknown:

Well, the I mean, the goal is to be a good human. Right? Like, I don't I don't think there's any way that there's not many people that will disagree   with being a good human. I don't think it's enough.  

[00:07:03] Unknown:

It's not enough to be a good human? No. I don't believe so. I I I don't think well, because you can't you just aren't being a good human isn't gonna put food in your mouth. You can't eat your good humanity. You know? This is why we have the issue with your that's why we have the issue with virtue signalers in the world. Well, when I say we go to human, I I would mean that in the context of being a productive member of society. Like, that's part of it. Right? There's there's this tribal aspect  

[00:07:27] Unknown:

of of humans.   Got you. And in the process of being a good human, math like math and cryptography is this tool   to essentially safeguard this is is is almost what what we've done. It's like,   it it's interesting because when as I go through these, like, cryptography stuff and this this math stuff,   I think a lot about the crypto wars of the nineties, and I don't mean crypto like the shit coins. The Philippines meant crypto wars. Exactly. The the PGP   print RSA on a t shirt crypto wars of the nineties when they try to   deem mathematics   as this weapon.  



[00:08:03] Unknown:

Yes.  

[00:08:05] Unknown:

And it and it's anything but. It's not it's not a it's not a weapon. And  

[00:08:10] Unknown:

it's hard to put words on this. Right? You know, you got all kinds of people that And we have people today casting Bitcoin as a weapon or Bitcoin as a aggressive weapon or something like that too. We Right. Going through very something similar.   Right. Right. Right. Right. I know all the feds have tweeted yesterday   something about the DOD.   You know, the the other feds when I say the feds, like the Michelle Weakley's, everybody who we call feds, who they get all butthurt when we call them feds, but then they post things like, hey. Guess what? The DOD is going to mine Bitcoin.  

[00:08:40] Unknown:

Yeah.   Sure.   I Like, don't But, like I I follow her almost immediately. We digress.  

[00:08:49] Unknown:

Sorry for the digression.  

[00:08:50] Unknown:

No. But, I mean, you're you're not wrong. Right? You know, I've I have close proximity to DC, so I'm interacting with, like, BPI folks. I'm interacting with the swamp creatures. Like, I'm, you know, going into DC. Yeah.   Not often, but bit DC Bit Devs. Right? Come come check out DC Bit Devs.   I'm not good. I'm one of those. Hell yeah. It's at MicroStrategy headquarters or strategy headquarters in Tysons Corner Of Virginia. There's also some other interesting buildings in the general vicinity of Tysons Corner. Do I need monetary security clearance to get into MicroStrategy's   headquarters? Do I need No. You don't. Nope. You just gotta say you're a Bitcoin. I hear from you, though. Not yet. Not yet.  

Right.  

[00:09:28] Unknown:

But let's let's jump in. Let's not, let's not So okay. So, you know, I think in the context of just educate you know, we're you know, look. We're both very involved in the education of our kids, and I think that it's a good thing for us to talk about because a lot of people who listen to this are, you know, very involved in the education of their kids and Yeah.   You know, hey, dude. I just saw you shaved your mustache.   You finally went full screen, and I'm like, damn. I knew there was something going on there.   Yep.  

I guess that's, you wanted to get taken seriously at the school.  

[00:10:04] Unknown:

Just, you know, in general.   My wife just texted me. Mustache, and we'll leave it at that. You don't wanna get turned away at Tyson's Corner for the bit devs. No. The first one I ripped was a mustache. It was great.  

[00:10:15] Unknown:

Okay. So so let me let me, we had something very so we had something very interesting, in my opinion, happen.   And to me, it's more of, like, validating why we do this podcast. And so we started pick you know, Dan was, like, the first hitchhiker we picked up.  

[00:10:30] Unknown:

Right?  

[00:10:32] Unknown:

Dan is, like, now he's, like, in the he's in our car quietly studying.   Okay.   Just, you know, every once in a while, he pops his head up. He's like, you guys are doing great. Thank you. Keep keep it going. Keep driving. And then we're like, yeah. Thank you. We need that big time. Yeah.   Well, we picked I I'd say we picked up another big hitchhiker along the way, and this is now   just kinda legendary   because it's not somebody who is trying to learn math. It's a guy, I think, who is way ahead of both of us,   who probably gets frustrated, who listens to this because,   I'm guessing he has a very sovereign you know, a desire to be sovereign.  

Probably gets frustrated at the level of math we talk about, and he has decided to be basically just boosted every single episode   and,  

[00:11:21] Unknown:

pretty substantially. And so we're we're going to re signal. Boost is like, the the it was, like, not just the sats were not just the value that were conveyed here. There was there was a lot more value conveyed. He also demonstrated some chops in the boosts because Absolutely. He got, he almost got, like, perfect LaTex notation  

[00:11:40] Unknown:

in the boost, which is really hard to do. Like, it's hard to get mathematical notation   in the boosts. And so we'll we'll, we're going to decipher   some of this notation, and I think like, I've read through them. I think I can do a decent justice into at least getting into  

[00:11:56] Unknown:

Alright. What he's doing. And If you if you struggle, I I I cheated and I asked the AI to talk to symbolize this. We're not doing that. No. We're not doing we're not bringing that in. Don't do that.   You know, don't do that because, you have to look up the name of something you don't how do you look up something, though, with without knowing the name?  

[00:12:13] Unknown:

I don't I I don't know. But, like, what I'm what the thing about AI here's what I'll say.   And,   hey, I can be I'm gonna give my speech how AI can be great. It's just that if you   it can rob you of your ability to learn.   Yeah. That's fine. That's all. And so   I wanna take this as an opportunity to,   to learn. And,   you know, eight Mythrandir is the man, and he is I think he has a lot to teach us. I think we have a lot to teach him. Okay?   And, he's he probably just got triggered by me saying that. But, let's, let's go through it. Okay?   So let's go back can we go well, I was gonna say we would go back to the very first episode   and   episode one   Yeah. And start with the boost there.  



[00:13:03] Unknown:

7,777  

[00:13:04] Unknown:

sats from eighth Why don't you hit that because it has no notation and you can you're capable of reading it.  

[00:13:09] Unknown:

Oh, thank you. Thank you. Well oh. I'm saying we're gonna hit some. We're gonna hit some that you're not you're not reading. You you just said I'm capable of reading it. Right? Because it's a language.   Yes. Like, that was a that was one of the huge things seeing this mathematical net. Anyway,   yeah, Ape Myth Randir,   legendary booster in the podcasting world. So thank you. But he goes,   Wolf Wolfram   math world was a great resource for us budding self taught self taught mathematicians   back in the day.   Yep. There's a lot in there. First out to Wolf Wolfram,  

[00:13:40] Unknown:

is yeah. He was a mathematician. I think he created   a a software called Mathematica, which   maybe some people still use.   Maybe There's, like, Wolfram Alpha. It's alpha too? Well, he basically figured out as, you know, he probably I would say, like, he kind of applied Moore's law to the for to the ability   to code math and to scale people accessing,   you know, being able to access   mathematical code.   I mean, I think Python is was such a   such a breakthrough that I don't know if anyone growing up today who already knows Python is gonna appreciate   what what what Wolfram   has done.  

But I would say people should check out Wolfram Alpha.   Yes.   All of the essentially Wolfram the world of Wolfram,   people should check that out. Wolfram math world?  

[00:14:33] Unknown:

Mathematica?   Absolutely. Yeah. These and these are, like, visualization tools and then also,   like, education like, definitions   and,   sort of like encyclopedia. Like, it's it's an encyclopedia. That's a good word for it. That's a good word for it.  

[00:14:51] Unknown:

So that's that was the first boost just right off the bat. First boost. So he basically just started I think he's like he meet I don't know what he saw. He saw that somebody in his world   I'm not that connected to him. I I wasn't really that connected to him outside of the podcast. I think we've maybe interacted once or twice.   But maybe somewhere in the intellectual Silk Road world, he saw we were doing a mad podcast, and he is like, I'm a mathematician.   Let me see what the hell this is. And Right. He listens to the first episode and doesn't just doesn't totally hate it. You know what?   But more which is more that I could say for Alan Farrington who listened to the first episode and just couldn't stop telling me what I did wrong. It was just really funny.  



[00:15:34] Unknown:

That's good. Yeah.  

[00:15:36] Unknown:

You know, he he didn't have that reaction, I guess. He just was more like, hey.   Check out Wolfram.   Well, in the next the next boost, though, he's Alan Alan didn't even boost us. You know? He just he just texted me. Don't boost shame people on this pod. It's okay. Well, I think it's okay.   As we say, we associate higher truth to the size of the boost. Right?   Now here's a good one. Here's a really good one. I'm a   Right. And this is insulting because I still don't know what these words mean. Yeah. So this is when I realized that this this boost is when I kinda realized that okay, this guy has some game and he also has some shit talking  

[00:16:15] Unknown:

game. Yeah. And,  

[00:16:17] Unknown:

he says   this is on episode two, exploring groups and mathematical structures.   He says Gary needs to lower his time preference and stop jumping ahead. He doesn't even know his homomorphisms   from his isomorphisms. Explanation point.   So What's your reaction to that?  

[00:16:38] Unknown:

What is What What is your reaction to that?  

[00:16:41] Unknown:

I At your at your level of understanding.  

[00:16:44] Unknown:

I still don't know what those mean. And and and when I read it, I I knew that I needed to probably go look it up, and I just didn't get to it,   before before today.  

[00:16:53] Unknown:

But he's probably be right. I'm just gonna assume because I don't know what they mean. My favorite subject, literally. It's literally the reason I study it. I study abstract algebra right now. Homomorphism   versus isomorphism is the Not versus more of just, yeah, versus and and   and, it's   so it's   I think I'm I'm gonna just take a wild guess because I remember episode two vividly. I listened to it probably seven times because it was so hard.   It was such a hard thing to do.   And the thing that I locked in on was this whole question is like, why do we care what a group is? Why the effing why do we care? Why do we just spend all this time? Right? And I think, like, by episode seven or eight, I think we figured out, like, why it's kinda cool and useful to know what a group is. And when we started when we started realizing that the points on an abelian curve I'm sorry. The points on an elliptic curve were an abelian group. And,   you know, it's also so there's, like, a isomorphism   between the points on an abelian curve   and the,   equivalence,   like, the basically, the equivalence class or the residue class of an integer,   like of a prime integer or a, you you know, the unit group of an integer,   we can say that the members of the unit group are also like, there's an isomorphism with the points in an elliptic curve. Why? Because they're both, if they have the same amount of elements and they and they're both abelian groups that you're now you're not collecting   you're not collecting,   properties   that you're starting to say, wow. These might be similar. I might be able to do similar math. I might be able to treat them similarly even though one is super confusing than elliptic curve and the other one is really simple. You see what I mean? So if you can basically take   two different things Say that. Why? Why? You can take two different things, one being super confusing but you care about operating on that.  



[00:18:42] Unknown:

Mhmm.  

[00:18:45] Unknown:

And then one that you actually understand you kinda really understand the properties of, which is like, you know, what we used to call what we were calling, like, the members of the finite field. Like, the group of,   you know, one through,   the prime minus one. All the like, that would contain all those numbers. Right.   So that's something we kinda understand. Wait. So you use isomorphism  

[00:19:06] Unknown:

again and and So isomorphism,  

[00:19:08] Unknown:

basically, what isomorphism   means is, like,   essentially what it's it's essentially,   an equality   between two   types of saying, like, two types of groups are equal. Like, the only difference is label is how you label is what you call them, is the linguistics.   So what you call the elements.   Right? What so you might call one the points on an elliptic curve, and you might call the others the integers.   Right? But if they have certain properties   now I don't wanna get into   I'm just gonna say what they are so that the math people can relax.  

[00:19:44] Unknown:

Sure.  

[00:19:45] Unknown:

Basically,   they,   they preserve   the same operation,   meaning,   and the operation is preserved under both. So, like,   f if you said f of,   a operates on b, it's, like, inside the parentheses that would equal f of a operating on f of b.   Good. That's just for the math people so you guys and then the other second thing is it's called the bijection, which means it's one to one and onto. I'm not gonna get into what that means. I just want the math people to understand that I kinda I know what an isomorphism is. Okay. So  

[00:20:21] Unknown:

isomorphism   is sort of the you're right. Like, some of the observations that we've made where the same sort of group theorems   and properties apply on this other thing. And they both share share a similar shape, but they are not the  

[00:20:36] Unknown:

same sort of like From the if you just here's the key. There's such a key right here. And if all you care about is that operation, and so with respect to the operation, it is they're the same.   If you just abstract everything else away,   you could just look at the two you could look at the two groups and say they're basically the same except for what we call the labeling.   The way they show this the way they show this in abstract algebra is like using, like a square   and,   like, the symmetries of a square if you rotate it.   The various options you have when the various outcomes of rotating and flipping a square and its axis   is a is a group   also.  

And it's like you can represent that group   using this whole the similar you you can represent it using something simpler that's less visual, but it's more of just the integers.   So I don't I don't think we're gonna get too derailed by that. I think Ape is laughing his ass off about how far we've gone afield on this thing.   That's right. This is probably the exactly what he was going for   is to have us read this and just lose our completely lose get derailed. And, I just wanna say for completion, a homomorphism   is an isomorphism without the bijection.  



[00:21:54] Unknown:

Meaning there's nothing else that it it sort of applies to? Meaning it apply well, it no.  

[00:21:59] Unknown:

You preserve the operation, but it's not like   it's not you can't it's not fully equal. It but it does preserve the operation.   So sorry about that for anyone who's, like, not a math person. This is, like, ape myth grandeur just   wrote this boost. We had to address it. But I oh,  

[00:22:17] Unknown:

well,  

[00:22:18] Unknown:

I think just he was thinking. I I think what he was thinking when he wrote the boost, though, was when I was saying that why do we care if something's grouped? And the answer is then it allows   us   to use   basically, it allows us to compare,   I think, two systems, and the math will be legit.   We we can rely on the math. And that's because of  

[00:22:40] Unknown:

isomorphism.  

[00:22:42] Unknown:

If something's an isomorphism, yes. Then we know that the the operations are gonna be preserved between the two, so we can do the same math there. We know the axioms are gonna be preserved. The   right? And we also know that the members will, like, under an isomorphism, we know that the members will exist under both.   Okay. So I I think we don't do much on there. Let's go to the next one. Now by the way, those are the easy ones. Do you want me to read this one aloud?   You know what? I think for, you know, let's give him let's give him his money's worth.   And,   I don't I  

[00:23:18] Unknown:

there okay. Alright. Let me let me give it let me let me attempt to read this out loud. Because it starts with mathematical notation, and I I did cheat. I looked some of these symbols up.   But it says,   for all   lowercase n's  

[00:23:31] Unknown:

So, like, you have okay. First symbol is an upside down a, which is the Right. Logo of Axiom,   capital. Alan Varangood's company, not to shout out him too much, but, like,  

[00:23:42] Unknown:

that is the Okay. But that's fine. What does it mean? Trade like, I don't care about the logo, what the symbol means. Logos are relevant to math. No. But it's the symbol. The symbol means  

[00:23:51] Unknown:

for all, for every single, for each.   And this is big. And it's five big single.  

[00:23:57] Unknown:

But but it has to, like, for all and then what and the the what is n, lowercase n,  

[00:24:03] Unknown:

within a set of uppercase n. So being an element of the natural numbers. So uppercase n is the set of natural numbers. And the set of natural numbers is all of, like, the positive integers. Yeah. And he specifies that down below. And it's a set of natural numbers. And non negative integers is probably a better way to say. And some people include zero sometimes in the natural numbers. Well, then there's this backwards e symbol.   And that means there exists. And so that's there exists   is,   like logically,   related to for each. It's almost like it's the logical inverse   of for each.  

Meaning, like,   one has a really strong condition. It has to be true for every single possible scenario in the world.   And the other one just says, no. It only has to be true once.  

[00:24:53] Unknown:

Well, it defines there exists p one comma p two ellipse dot dot dot comma  

[00:24:59] Unknown:

p k. Ps are primes. I'm guessing p means prime. Well, yeah, he specifies it. Distinct prime numbers. P one two through p k are distinct prime numbers. This is gonna be our worst episode, but And We got paid to do it. We gotta do it. Well, go look at the boost. Like, if you're listening right now on fountain Yes. I have it. Like, pause it. Go look at this the boost,  

[00:25:15] Unknown:

I have a yeah. Pause it. Go look at this the boost on on motivating the math zero three chapter two modular arithmetic, the queen b for maybe three years. Nostril post. Maybe I'll put I'll create a nostril post that   aggregates these boosts. That's not gonna be my goal. Sure. Yeah. Yeah. But but either way, you can you can follow along if you do that. Right? So for all n,   within   capital n as natural numbers   Yeah. There's this backwards e, which stands for there exists   distinct prime numbers, p one, p two,   etcetera, etcetera, up to p k   within the set of   prime numbers, capital p.  



[00:25:54] Unknown:

Yep.  

[00:25:55] Unknown:

There and then then and then there's another backwards e, which so there also exists   Yeah. Eone, e 2 Dot through e k where e   are the respective   powers,   non negative integers of those primes.  

[00:26:12] Unknown:

Yeah.  

[00:26:14] Unknown:

Well, yeah. That's right. So In in the natural numbers,   there exists within natural numbers Yep.   Such that and then and then this is gonna get tricky where it's   n equals   p one to the   e one power   times p two to the e two power and so on and so forth until you get to p k to the e k power.  

[00:26:40] Unknown:

Yeah. Just decided I'm gonna do a video on this. But, I mean, let me sum it up. Okay. This is called prime this is called prime power factorization, and there is a theorem   that says that every single number   on earth   or or otherwise, every single number that can be conjured in the human brain or maybe even or maybe even not. Every number every   natural number   can be represented   as the product   of primes  

[00:27:07] Unknown:

or their prime powers, really prime powers. And and elaborate on prime powers. Like, what do you what do you mean a prime power? Okay. So listen. So we have,  

[00:27:17] Unknown:

six is two times three. Those those are primes. Right? Mhmm.   The number seven is one times seven.   Mhmm.   But the number,   12   is   four times three. So that's two squared times three. You see? So two now you have p one is two, e one would be two. Two to the second power times three is 12.   So in other words, you can always   represent the number by some product   of primes with their powers.  

[00:27:54] Unknown:

And so 12 could be represent so, yeah, 12 represented as two to the second power   with three to the first power. Right? That's right.   Interesting.  

[00:28:07] Unknown:

You see so, so What is the k where's the k like, what the 50 is two squared times five squared. Right? K is just the last number. That's just all all they're saying is that there's always a series of numbers. K could be one,   I e,  

[00:28:23] Unknown:

four is two squared. Correct the record. 50 is two to the first times five squared.  

[00:28:28] Unknown:

Sorry.   Thank you.   Yeah. So 100 would be two squared times. That's,   boy, good catch, man.   Usually, we gotta wait till the end of the episode and It felt good. We cut it out.   Okay. Great. So, yes, so that's this is a theorem, and, you know, I think I've seen it called the prime number theorem or the prime power factorization   theorem   that, and this is   this is huge, by the way. It's one of the most important things.   It's huge. It's really, really important.   It's actually a really important theorem. So I think I I don't know if he's trying to tell us something really important or if he's just trying to fuck with us. But either way, this is a really important   theorem. And   if you remember   if you remember when we did the first episode, I intended to go through prime numbers. I was gonna go I was going to do this theorem, but not with this notation.  



[00:29:24] Unknown:

I mean, is this theorem important?   Because,   I I try to think back to so, like, prime numbers. Right? Like, what, like, what makes them prime? It's like themselves in one. Right? So they, like, they they live in this space where there's just themselves in one   as sort of the the factors. Right?   Yep.   And so if you can bubble any number down and maybe bubble is not the right word there, but it's kinda in my mind. Reduce there you go. Yeah. The correct word is reduce because  

[00:29:55] Unknown:

it's it's close to irreducibility.  

[00:29:57] Unknown:

Well, when you reduce something on the stove, it's gonna bubble a little bit. So keep waiting for me here. But reduce, you can reduce it down to these things called primes that have these very, very special properties.  

[00:30:08] Unknown:

And then you can get to a version where you know you you know you can no longer reduce   the number.   Irreducible.   Yeah. Correct. At least in with with not not with natural numbers.  

[00:30:21] Unknown:

Right.   Okay. And that's interesting that's interesting that natural numbers like, this is a property of natural numbers too, just like that terminology. Like, who who used the word natural   to describe these numbers?  

[00:30:34] Unknown:

Good question.   I'm not sure I know the answer to that one,   but,   I think that was the first number system people thought of is my guess. And,   then they needed to extend it to the integers. They need it because they needed an inverse.   Right? They needed to solve for a zero, which you couldn't you can't do with the natural numbers.  

[00:30:57] Unknown:

I don't know if we'll ever know for certain on that one. Well, I'm sure because sure because where did the numbers come from to begin with? Right? Like, that's Somebody That's another  

[00:31:06] Unknown:

No. I mean, I think it's fairly somebody knows who coined these terms. I just I'm not an a student enough in math history. I haven't studied enough math history to know for sure, but I have some pretty sure somebody knows the answer to  

[00:31:18] Unknown:

that. Okay. But, like, the numbers themselves. Right? So there's   back to, like,   math and numbers being a language. Right? Like, the where did the numbers come from? You know what I mean?   Like, is it just because we have ten ten fingers and 10 toes is base 10, but base 10 is just one base that makes it convenient for us as monkeys to, like, do all this math. Yeah. But that is why. Yes. But that is why we use base 10 in our culture.   Yeah. But then there's all these other numbers of different bases and properties that you get there in. I understand. So Maybe we'll get into that. It's the same   question. Like, where did English come from? Where did language? Like, just language in general, not just English. Like, but where did language come from? And I don't know if there's a   a   non supernatural  

[00:32:02] Unknown:

explanation for that. True. And I don't know if there's anything in language that formalizes   it the way rings do,   you know, the way rings do for polynomials and for numbers. I don't think that I don't know if there's such an analog in in language.   I mean, I think there's I think that's where grammar comes in. Right? Like, it's it's the grammar, the structure of it. I don't know if it's as elegant as I don't you know what I mean? Maybe for some languages it is, but not for English.  

[00:32:29] Unknown:

It's my guess. Well, you can you can formulaically   define. Right? So, like, one of the   most common things is, like, the farmer is great. And in Latin, that's Agricola est magnus.   Right? So, like, there's the but it's like an like, that is   on either side. You can have two things and, like, in language, you're comparing those things. It's like a comparison sort of thing or like a,   I don't wanna say equivalent, but it sounds like a word we use for, like, math.  

[00:32:55] Unknown:

Maybe maybe Latin is, like, the base two where we can put everything in there and it makes sense out of it.  

[00:33:01] Unknown:

A lot a lot of English does. Right? A lot of the romance languages do because they they they tend to have derived from Latin.   But that's a whole that's like a language podcast,   which I guess we are a language podcast. Right? I think we are in some sense.  

[00:33:19] Unknown:

So let's let's hit the next one. Yeah. Yeah. Okay.   So we are continuing on eight with Mithrandir's boosting spree.  

[00:33:26] Unknown:

He was able to get the correct,   n symbol here for natural numbers now. Really did. This dude is marathoning.  

[00:33:33] Unknown:

He's marathoning this thing. Yeah.   So we're now in episode four, which was the one where we went through the geometric series and the Bitcoin supply.   Mhmm.   I don't know I don't know if that's relevant to this, but I'm I'll read this one out. He says for all n in the natural numbers,   there exists   a natural number k.   Right? And   some a, we call he calls it a I, but not like artificial intelligence. A underscore I. Not like Allen Iverson.   It's   a subscript I,   which that's gonna be in the finite field   with the prime b, meaning it's gonna be in that it's gonna be either zero, one, dot dot dot b minus one. K. Right? So we're we're looking at two numbers, the k that could be any natural number, and a subscript I   that   is in this finite field with a prime of b.  

K?   Mhmm. Now so there exists these two numbers such that if I sum   this a I to the power of b I,   I get n.   Woah.   Well,   nothing   super clicking here except   for the fat fat. What also specifies that b is greater than one? So here's the thing. First things first.   Yes. B clearly has to be greater than one. Otherwise, there's no field. Right? Yeah.   What I think   what what I would think here   is that,   you have a cyclic group.   See that these member AI is part of a sick they're saying, like, it's it's in a,   the group basically, like, the group modulo,   b,   and b is prime. So every number zero, one, etcetera,   is in the group. It's also it's all it's relatively prime   to b. So it's like the unit group of b. I don't know if it has to be prime, though. Right? Because b was 10. B has to then we we would then I'm guessing this would be true in the unit group,   but it's easier to express   zero one dot dot dot b minus one.  

We know that we know that b if b is a prime, that'll be true. We know that we know that all those members will be in that. Like, the members of the field, modulo b, will be in that group.   It's okay. Every number every number less than b will be in there. So this is the one of the ones I had to paste I pasted in because I I just Hold on. Let me let me let me keep running. Yeah. Let me keep trying to figure this out. So then Okay. So n   can always   so every number. This is kinda like the prime this is maybe a corollary to the prime power   theorem   from before.  

Right? Mhmm. Maybe like a core maybe a corollary, maybe not. Ape is loving me struggling through this. So I really wanna give this guy his money's worth. This is a lot of a lot of sats.   So so then you can always say that the sum of AI   to the BI   Maybe this is just base this is a form of base notation too   that that right? It's just this is just true that you can,   a is the base.   Sorry. B is the base. And so you cannot and and I think that's what it is. Yes. I think that's what this is. Well,  

[00:36:55] Unknown:

again, according to some random,   AI site that perplexity.ai   that I plugged this into, that was that's what it spit out.   And it makes sense because if b is is 10   Yep. You have zero through nine, b minus one,   for the for the numbers a I Yeah.   Such that as sum of   go ahead.  

[00:37:18] Unknown:

I think we've said this in the podcast before that,   every numb remember we said once that every digit, every number,   not every digit, but every number is actually a polynomial?   Right. This he must be responding to that. This is this is his code to tell us that he hears me. And, you know,  

[00:37:40] Unknown:

I don't know. It's reduced to yeah. You can reduce it to a polynomial. So every number is a polynomial  

[00:37:45] Unknown:

where the coefficients remember we said the coefficients   Yep. Are   in the literature, they'll say they're in the ring.   Okay? They're in a certain ring, the coefficients,   because the coefficients could be polynomials.   But when the coefficients are actually natural numbers  

[00:38:00] Unknown:

Which that's what it's talking about here. What k is.  

[00:38:04] Unknown:

Right?   K,   sorry. A I is the coefficients   here. I don't wanna get this wrong.   A sub I are the coefficients and they're in this field. Field is also a ring, but we usually talk about polynomials   in context of a ring. So a I is a field. Mhmm. That defines the coefficients. So like you said, in base 10, it would be zero through nine,   And then b to the i's would be 10 to the whatever.   Right? 10 to the something. And we sum it all up. And that's how every number is actually a polynomial, and I think that's why he boosted this.  

[00:38:40] Unknown:

You know? K is is what is k represented here? Right? It's so k is a number   there is a a natural number.  

[00:38:50] Unknown:

Yeah. So k is the number of digits in the number. K is just saying it's a natural number. Like, that's Well, no. That the number of digits is a natural number. In other words, you can't sign over 6.25   digits.   Ah, yeah. Okay. That makes sense. Yeah. Yeah. To be just yeah. It's that's a formality.   It's a formality of the proof. Bounding  

[00:39:13] Unknown:

how,   it it's like bounding a and b in, like, in the iterations. Right? So, like, when I is zero because it's I equals zero two k.   So you start with zero. So   a when a zero times   10 to the zeroth power is just one.   And then   I and and k is a natural number, and it goes up to at least in base 10, it would go up to nine.   It'd be zero, one, two, three, four, five, six, seven, eight, nine.   Yeah. Okay. That makes sense.  

[00:39:45] Unknown:

Alright. Let's keep going.   You wanna read this one? You may or may not be able to read this one. There's no there's no symbols, though. There's  

[00:39:57] Unknown:

yeah. So,   episode five, elliptic curves and Fermat's Little Theorem.   Mabe says, I feel like you were very close to explaining the sieve of aratosthenes.   Aratosthenes? May I  

[00:40:10] Unknown:

have might be I was   I'm I'm a sneaky bastard because Eredo I couldn't have tried to thanes? I I would have thought of arith Aristothenes.   That's how I think I would have said it, but I don't know the answer, honestly. E r a t o s t h e n e s.  

[00:40:28] Unknown:

There's a sieve that this person has multiple times. And we get close to explaining the sieve   of Erathos Aristosthenes.   Aristosthenes  

[00:40:36] Unknown:

multiple times during this podcast, but never quite got there. Yeah. So now I completely I I remember distinctly, I think, why   why he boosted this. I   I was saying that if you wanna find the fact is this is I was saying if you wanna factorize a number   that,   you just have to do it by hand.   So for example, if you wanted to factorize the number a hundred like, you wanted to see if a hundred and one was prime or if it had or if it was before it had factors.   Mhmm. You could go up to the square root of a hundred and one, the integer of that, which is 10. Right? Square root of a hundred and one. The integer of that is 10.   And then you can basically ask if each one is a factor, which is   like very similar to this. The sieve of Aristosthenes   I'm I'm gonna pronounce it that way. There are a number of there are a number of what's called sieve methods to find prime to find prime numbers. And these are like like Kablets writes a lot about them in his in his book.  

It's very brute force to go one by one and ask if this is it's a very hard thing to do. Primality testing is very difficult. So you need algorithms, you need tools, you need tricks.   Okay. And with the sea of Aristastin, he's done it's like an old school primitive way of just   basically writing out all of the prime all of the numbers. Right? So imagine writing out every number, say, from a one to a hundred. Mhmm. And then you take you know, two is not two is prime, but anything so two is the first prime. And so you can take any number   that   two divides, which I eat every even number and just cross it out.  

K. Right?   So you give your list of numbers to a hundred and you just crossed out you circle one, you circle two because those are prime, and then you cross out every even number after that.   Right?   Right. You cross out four, six, 10, etcetera, up to 100.   Then you have,   about,   what, 48 numbers left? Mhmm. Then you say you go to the next number and you say three. Okay.   So three is clearly prime because   I haven't crossed it out. It's the lowest number I haven't crossed out, so it's prime.   And then I can cross out every number divisible by three. So you hit nine, you cross off, you know, you cross off nine, you cross off 15.   Mhmm. Twelve twelve is already crossed off. Six is already crossed off because they were even. So then you cross   off all those odd numbers, and now you're left with,   you know,   I don't know, something's you know, some less than number. And then you say, okay. What's the next number I haven't crossed off? It's five. Okay. Now I'm gonna cross off fifth well, I already crossed off 10 and 15, but I'm gonna cross off 25,   30 5.  

Right?   45 is gonna be crossed off. So if you wanna cross off 55,   right,   65,   except, you know, 85, 90 5. I'm gonna cross those off. Then I'm gonna go to the next number, which is seven.   I'm gonna circle that, and I'm gonna cross off any number in multiple of seven. This is what the sieve of arystathons in these does. And then you find that you're left with   from, you know, you're left with   the prime numbers. You're left with the numbers that haven't been crossed off. Once you hit   basically 10,   right, which is square root of a hundred, you there's no more numbers you can cross off.   That's what and so it's a very similar,   it's like a similar idea   to what I was saying. I mean, finding a prime finding the prime numbers   is a   like, that's a,   universally   valid thing that a lot of people wanna do. Figuring out if a number's composite is downstream.  

Right? Because if you know what the prime numbers are, then   you know if a hundred and one is composite if you've done the sweep of arystosthenes. You don't have to  

[00:44:21] Unknown:

Yeah. It it's and and it I it's I I think it's a lovely,   word to use, sieve, because it it sort of illustrates. Like, you put all these things into it, and then you get out what you want. Exactly. You drop you dump every number into it, and you hope the ones you want pop, like, stick. Right?   Yeah.  

[00:44:40] Unknown:

Yeah. And, actually, so if you look up, there's a lot of c's. I mean, I'm guessing   it's an entire   it's probably an entire number theory course or entire cryptography course   just related to all of the sieves that are used in primality testing.  

[00:44:56] Unknown:

I would and I bet that   I bet   math world will have them listed somewhere. Primality yeah. There's a whole I would get through that. Yeah. So I'm just I'm browsing this this sieve of air   Eratosthenes?   Eratosthenes.  

[00:45:13] Unknown:

Thank you. Off the tongue, so I'm going with it. Yeah. Eratosthenes.  

[00:45:17] Unknown:

And it it's just like the subcategories.   Right? So, like,   for the listeners because there's no viewers,   there are, like, a bunch of subcategories. So under Siva Verastases, like, top level one is number theory, special numbers, sieve related numbers.   There's also number theory, prime numbers, primality testing,   which that primality testing you were just talking about. And here's a whole list of   of,  

[00:45:41] Unknown:

I just see one sieve, though. So how hard is your hard stops? This is the question.  

[00:45:46] Unknown:

I think it's probably a good time to cut here. I I don't think we have time to cover another boost.  

[00:45:51] Unknown:

No?   No.   Really? This is One more?  

[00:45:57] Unknown:

Yeah. We can cover the next one.  

[00:46:00] Unknown:

We'll cover the next one maybe not so great. I have a pretty I think I have an answer for you on this one.  

[00:46:05] Unknown:

Yes. Okay.  

[00:46:07] Unknown:

By the way, you guys wanna really just   derail a full episode of our podcast. Just send 10 boosts with   really   challenging math, and we'll figure it out on the show.   Okay. I'll read this one just for brevity.  

[00:46:21] Unknown:

Sure.  

[00:46:22] Unknown:

And maybe I won't read the whole thing out. Basically, it's saying a and b are natural numbers with b greater than a greater than zero. Oh, you skipped one.   Did I?  

[00:46:32] Unknown:

Yeah. I was going off of the list on the on the screen.  

[00:46:35] Unknown:

Oh, I just I'm going episode by episode. So this is now episode six.   Send it.   You want me to just yeah. I'm sure I'll just keep going. It's very similar. And I mean, actually, it's a similar similar spirit.   It's a it's a those two are very similar spirit.   Maybe I'll combine them here.   Abe's like, no, man. I gave you a separate boost. You'll do them separately. It's   kinda ugly, but it turns out the sum function is   the sum of,   terms of Fibonacci   sequence.   Okay. And it's an alternating series. So it's negative one to the power of n, big n minus one   times   Fibonacci of big the nth big nth Fibonacci   plus negative one to the big n   times   f of big n minus one.  

Sorry. Sorry. Sorry. Sorry. Oh my god. We're fucking destroying destroying this by having one. Should've cut it. Let's, let's rip this one next time because I don't know if we're gonna be able to close this one out. Right. Keep the struggle in. Keep the struggle in, but, let's you wanna have any okay. Let's leave it here. Let's leave it here. K? You wanna have any   you wanna have any close   a slight want any kind of closing conversation to try to add some value   outside of outside of this struggle to up to anyone other than Abram with Randere?   Thank you, buddy.  



[00:48:03] Unknown:

No. I think I think one of the biggest things that this boost highlighted for me,   was, like, I I don't think I'm gonna get much farther if I don't understand these annotations.   The these,   like, the the language of math. Right? Like, these symbols that mean, oh, the n when it's drawn this certain way stands for natural numbers.   Yeah. And so I think that's, like, that's that's a deficit that I have that I'm gonna have to build up. It's true. But I don't so  

[00:48:30] Unknown:

I would say this to everybody here. Okay? We had a lot of fun doing this.   Yeah.   But   I wouldn't get so one of the reasons   the reason I do this podcast, and I think the reason we both do it here together, is we're trying to demystify   trying to demystify   a lot of what goes on in math. And I think I think a lot of people are really intimidated by things, thinking they're really hard when   all it is is that you're just not used to notation   in a lot of cases. And,   like, what we did with groups   in episode two   was a pain in the ass.   But   look at I you know, I talked about this last week. Right? All of a sudden, Gary is, like, using   group concepts   very fluidly because we went through it.  

See, all we had to do is demystify   what is going on and spend a little bit of time on it.   Right?   A little bit of iteration, a little bit of repetition.   And then, you know, within a couple of weeks, you're actually all of a sudden, you get it.   And so   I wouldn't again, I, like,   I don't I don't know if Abe is trying to demoralize anybody here. I don't think he is. Right? Not at all. I don't think so. If anything, he's testing me.   I think that may be. Right? I think because people do that, and they should. You should test me. Test me. See, you know, do it. Like, I don't mean testing me to see where I'm where I rise up, see where when see where my limits are. Right? I'm pretty open about it. That's the whole Go ahead. That's the whole thing we're dancing here. Right? Is is the test of how far  

[00:50:13] Unknown:

how far can we go with our understanding,   and can we get to the point Freeze. Where that   I think it might have just frozen up. I did. I yeah. I did just freeze.   But but it's it's the test to see how far we can take this. Right? And and that's the whole thing. And 1%   better every time is is the the the low bar to hit.  

[00:50:37] Unknown:

So so I'd really wanna say don't be intimidated.   Look at the notation and   do do me a favor. Say to yourself,   I I don't know what the hell this means, but I will. This is nothing. It's just a bunch of letters. It all means something.   All I need to know is the translation and get used to it. And there's a math there's a famous mathematician named Von Neumann who had a great quote,   and you can cut it after this quote perfectly.   He said famously,   you never really,   I'm gonna butcher it. But he basically said you never really you never really understand math. You just get used to it.  

[00:51:12] Unknown:

Oh, I mean, that's everything. That's suffering in life. So suffer more.