- 5/18/2025
exploring bezout world
Category
📚
LearningTranscript
00:00:00Oh
00:00:04Hello
00:00:07Welcome to Friday Night Live with Hunky Lee episode
00:00:13208.4 yeah happy
00:00:17Sunday morning, it's like 1130
00:00:21Sunday morning. Yeah, I woke up and
00:00:25Brush my teeth and
00:00:28Went outside and did some running around the neighbor good jogging running and also walking. Yeah running and walking
00:00:35Okay
00:00:37now let's
00:00:39We get back to mathematics. Okay, so I
00:00:42kind of thought about that problem like that kind of like a basic coefficient problem and
00:00:49Yeah, I say well, I think I came up with some
00:00:53Algorithm maybe not the same algorithm as we discovered half a year ago perhaps better
00:01:00More efficient. I don't know. We'll see. Okay. Yeah. Yeah, so we kind of solve the problem
00:01:05Okay, so I show that with it better in this episode and I was like, you know, let's not be result oriented
00:01:12Let's be more
00:01:14fundamentalist purist
00:01:16For a while a list and get back to the basics and understand what's going on as opposed to like
00:01:23Hunting for the pattern, you know, yeah, let's just get back to the basic and understand what's going on
00:01:29Okay, and then I got some good results. Okay. Yeah, I did not write down anything on my notebook. I
00:01:35have notebook and pencil in my
00:01:38Bedroom on my bed, but I didn't use it. I just
00:01:42I worked it out in my head. Okay. Yeah, so
00:01:48I'm decent at mathematics right now. Okay, so I share that with you
00:01:53It's kind of like it's one particular generalization of Bezoo
00:01:58Coefficient theorem. Okay, so it's broader more general more powerful than
00:02:04Bezoo coefficient theorem. Okay. So yeah, so yeah, I'll show that with you. Okay, so now let's do stretching. Okay. Yeah
00:02:14Happy Sunday morning
00:02:20Okay stretching
00:02:23All right
00:02:29Yeah, I ran outside came back and open all the windows
00:02:34You
00:02:47Yeah, whether it's overcast meaning is cloudy it's nice
00:03:03You
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00:04:03Silence.
00:04:21Silence.
00:04:49All right.
00:05:03Oh.
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00:08:28Okay.
00:08:30All right.
00:08:32Exercise.
00:08:57Okay.
00:09:00Mm-hmm.
00:09:05No.
00:09:20All right.
00:09:21Let's see.
00:09:33Uh, let's do jumping kicks, okay?
00:09:38Yeah.
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00:09:51Okay.
00:09:52Good.
00:09:53Nice.
00:09:54Five minutes break.
00:09:56Good exercise.
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00:14:51Yeah, welcome.
00:14:53Happy Sunday morning.
00:14:55Thank you for joining us.
00:14:58Let me get some water.
00:15:20Yeah, so in human rights school, we respect all the religions, okay?
00:15:24It just happens that I happen to grow up as a Christian, although I'm a humanologist now.
00:15:31So, yeah, Mr. Jesus said, like, those who are pure at heart should see the face of God.
00:15:38Yeah, very good poetic lesson there.
00:15:41So, yeah, when we do mathematics, we are very purest.
00:15:45So, and we see the mathematical truth.
00:15:50We discover that, right?
00:15:51Yeah.
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00:16:58Good.
00:16:59Yeah.
00:17:08Yeah.
00:17:10So we're just doing this to your
00:17:17Okay.
00:17:18Ah.
00:17:19Okay.
00:17:20Yeah.
00:17:21Looks like some kind of number theory, okay, but I just don't remember what this is about,
00:17:45but it's already recorded in Human RSSeries, so we have to move on, okay?
00:17:50Yeah, we'll just erase it, because we need whiteboard, okay?
00:17:54So, and, yeah.
00:17:55All right.
00:17:56Okay.
00:17:57Yeah.
00:17:58All right.
00:17:59So, we're going to name this new theorem in number theory, it's a new theorem in number
00:18:26theory, Integral Linear Combination Theorem, okay?
00:18:34Integral, that sounds more like calculus, we are not doing calculus, but some mathematicians
00:18:43use that adjective as to mean like integer, but we are not going to, we don't want to
00:18:50sound like calculus, because we are not doing calculus, we are doing number theory, okay?
00:18:54No.
00:18:55Although some number theorists, some number theory, there's some application of calculus,
00:19:00but we are not doing that now, okay?
00:19:03Cheers.
00:19:04It's like generalization of Bayes' coefficient theorem, okay?
00:19:12What is Bayes' coefficient theorem?
00:19:15Common divisor of A and B can be expressed as linear combination of A and B, okay?
00:19:22But I thought about that in my bed, and then actually we can generalize that theorem, okay?
00:19:32And we call that theorem, Integral Linear Combination Theorem, and we'll prove it, okay?
00:19:42And the statement is this.
00:19:56Integral Linear Combination Theorem, okay?
00:20:08Let's put some title at the right or far right corner, okay?
00:20:12So that I remember when I see this whiteboard, okay?
00:20:15Although it's acronym, but better than nothing.
00:20:21Cheers.
00:20:26So let me just explain in plain English, okay?
00:20:30So, gosh, I'm so hungry, but yeah.
00:20:46I start cooking breakfast soon, well, lunch soon.
00:20:50It's like, the motivation is from our previous example, okay, Bayes' 5, where...
00:21:10Let's switch the whiteboard, okay?
00:21:19Okay?
00:21:30Okay?
00:21:31So here we are expressing 1 as linear combination of 1 and 5, 2 and 5, 3 and 5, 4 and 5, right?
00:21:48But if we can express 1 in terms of 1 and 5 or 2 and 5, that means we can express any integer in terms of linear combination of 2 and 5, 1 and 5, okay?
00:22:10Because let's say 1 and 5 is like 1 times 1 plus 5 times 0, right?
00:22:21And when it equals 2 and 5, 1 is equal to 2 times minus 2 plus 5 times 1, all right?
00:22:35So that equals to 1.
00:22:36But if we multiply that number 1 to any integer, it becomes that number, okay?
00:22:44So, well, now that we have some room here.
00:22:57It's an easy concept, but also very interesting.
00:23:041 is equal to...
00:23:07We express 1 in terms of linear combination of 2 and 5, okay?
00:23:122 times minus 2 plus 5 times 1.
00:23:23Plus, okay?
00:23:241 is equal to minus 4 plus 5.
00:23:28Then integer n is 1...
00:23:32Integer k, arbitrary integer k, is equal to 1 times k, which is 2 times minus 2k plus 5 times k.
00:23:46So any integer k can be expressed as linear combination involving 2 and 5, okay?
00:23:58Not just 1, not just greatest common divisor, no.
00:24:03Yes, Bez is right.
00:24:07We can express greatest common divisor of 2 and 5, which is 1, in terms of linear combination of 2 and 5.
00:24:16But we are doing more than that now.
00:24:18Any integer, not just 1, but any integer can be expressed as linear combination of 2 and 5.
00:24:28This is something really cool.
00:24:30Okay?
00:24:33Yeah.
00:24:40And it's not just 2 and 5.
00:24:43And it's not just 2 and 5.
00:24:50I'll give you a statement without writing anything down, verbally.
00:24:55If you want to, think about it, okay?
00:24:58We have two integers.
00:25:01It could be odd number and odd number.
00:25:04Or it could be odd number and even number.
00:25:07Or even number and odd number.
00:25:09Order doesn't matter, okay?
00:25:10Or it could be even number and even number.
00:25:13Okay?
00:25:15So when it comes to the pair of numbers, odd and odd, or odd and even,
00:25:21in those cases, these two numbers, a and b, okay?
00:25:26By some linear combination of these two numbers, we can generate every single integer.
00:25:33Okay?
00:25:34When it comes to even number and even number,
00:25:38we can generate any kind of even integer as a linear combination of two even numbers.
00:25:46Okay?
00:25:47And given that, we can come up with algorithm to find those coefficients.
00:25:58Okay?
00:25:59So we're going to do step by step.
00:26:02Is this publication material?
00:26:04Absolutely it is.
00:26:05Okay, yeah.
00:26:08Okay.
00:26:10Now, let's take five minutes break, and I'm going to start cooking my lunch.
00:26:16Okay?
00:26:18Okay.
00:26:19Yeah, five minutes break, okay?
00:26:21Yeah, if you want to think about that, even prove it,
00:26:25integer linear combination conjecture now
00:26:29soon to be, it will be theorem, because we're going to prove it.
00:26:33Yeah, if you want to prove it, go for it, okay?
00:26:35Yeah.
00:26:37If you want to disprove it, yeah, go for it too, okay?
00:26:40But most likely prove, okay?
00:26:42Yeah.
00:26:43All right, five minutes, thank you.
00:26:45Yep.
00:26:47Okay.
00:26:59All right.
00:33:56So we are working on this.
00:34:03Actually I need to qualify this statement a little bit.
00:34:10Because I gave those examples already.
00:34:15So let's, before we make this conjectural statement, okay?
00:34:22Let's make some concrete example, okay?
00:34:26We deal with two odd integers, positive odd numbers,
00:34:32that's not equal to each other, okay?
00:34:35Like three and five, not three and three, okay?
00:34:38So some two odd integers, okay?
00:34:43And let's make, well, easier example.
00:34:51Let's think about non-zero even number.
00:34:54Non-zero, let's think of a positive even number, okay?
00:35:02And an odd number, positive odd number, okay?
00:35:06And with some linear combination, we'll make one, okay?
00:35:17Let's start from there.
00:35:22Oh, so the problem is this, okay?
00:35:27Proof, just a moment, okay?
00:35:44There exists, this e like right, side, left,
00:35:48you swapped, okay?
00:35:50Proof, there exists integer x comma y,
00:35:58okay?
00:36:00So parenthesis, okay, meaning that this exists operator,
00:36:05this not binary operator, it's like unary operator, okay?
00:36:13And this member, set member function,
00:36:18it is also unary operator, so yeah,
00:36:22this exist and belong to integer,
00:36:25it also applies both x and y,
00:36:27that's why we have parenthesis,
00:36:29it's like a distributive law, okay?
00:36:32Brand new notation coming in there, okay?
00:36:36There exists x and y, also future integers, okay?
00:36:43Such that, and also, oh, such that,
00:36:55x, y, m, n, okay?
00:37:05And this pen is wearing out,
00:37:11but should be good enough for now.
00:37:13No, I want a pen that is not wearing out, okay?
00:37:26Yeah, this one's good.
00:37:33There exists, oh.
00:37:36Oh.
00:37:52There exists x and y, they're all integers,
00:37:57such that, for all integers,
00:38:05integer, m, n,
00:38:20we're dealing with odd number and even number, okay?
00:38:23So, two m, so x and y, they are coefficients, okay?
00:38:28So, x and y, they are coefficients, okay?
00:38:31Plus two n plus one, so we have even number
00:38:38and odd number, is equal to one, okay?
00:38:48That's a good start, it's a conjunction, okay?
00:38:51So, our job is to prove this.
00:38:56Cheers.
00:38:58Cheers.
00:39:09Whoa.
00:39:25Okay.
00:39:29So, one is an odd number, two m, even number, right?
00:39:35Two m plus one, odd number.
00:39:37So, y has to be odd number, okay?
00:39:41Why?
00:39:42Because even number plus even number
00:39:46would be even number, right?
00:39:47So, but one is odd number,
00:39:49that means this has to be odd number.
00:39:53And now, we have odd number times odd number.
00:39:57Is equal to odd number.
00:39:59That's why y has to be odd number,
00:40:00because odd number times even number
00:40:02would be even number, okay?
00:40:03So, y has to be odd number, okay?
00:40:07So, now, here's proof.
00:40:14So, y is odd integer, okay?
00:40:25Yeah.
00:40:27Yeah.
00:40:29Then, y is equal to two k plus one, odd number, okay?
00:40:42K being any integer, okay?
00:40:47Now, you want to finish this proof?
00:40:50Go for it, okay?
00:40:52You want to disprove it?
00:40:53Go for it, too.
00:40:54We'll take five minutes break, okay?
00:40:56Let me take off my lunch.
00:40:58Cheers.
00:40:59We'll take it slow, okay?
00:41:00Yeah.
00:41:04I'm unemployed.
00:41:05I'm between jobs.
00:41:07And today happens to be Sunday, okay?
00:41:11We have all the time we need.
00:41:13Okay?
00:41:14Yeah.
00:41:15Well, I applied to many jobs.
00:41:17More than 2,000 jobs.
00:41:18So, yeah, it's waiting game.
00:41:22Yeah.
00:41:23It's so cool.
00:41:26Yeah.
00:41:45All right, five minutes.
00:41:47Yep.
00:41:51Yeah, very exciting.
00:41:56Yeah.
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00:48:07Yeah.
00:48:08Yeah.
00:48:09So let's have a brief, small break from mathematics.
00:48:10Yeah.
00:48:11In Southeast Asia, wonderful gentleman who migrated to America came up with this Sriracha
00:48:19sauce, right?
00:48:20right? He's either from like Vietnam or Thailand somewhere,
00:48:25okay, Southeast Asia, Sriracha sauce, legendary. And then they
00:48:31come up with the ramen noodle based on Sriracha sauce. So
00:48:34I guess that's what I'm cooking. Okay, so the chicken,
00:48:37corn and the cabbages. Yes. Okay. So yeah, it's kind of all
00:48:47moral of the story is this, you don't have to be advanced
00:48:50mathematician to find, to discover brand new theorem in
00:48:54mathematics. Okay, look at me. My mathematics is not that
00:48:56advanced. Advanced mathematics is not bad, but not too much.
00:49:01Okay. What we are dealing with here is like middle school,
00:49:04high school algebra. Okay. Yeah. We are not dealing with
00:49:07calculus or complex number, nothing fancy like that. Okay.
00:49:14Is this brand new? I think so. Okay. Yeah. I don't know. I
00:49:20haven't seen anything like this before. So most likely it's
00:49:24brand new. Okay. Yeah.
00:49:27But who knows? Maybe somebody already discovered this. I
00:49:31don't know.
00:49:34Let's continue. Okay. Did you already prove this? Well, good
00:49:46for you. Okay. If so. Alright. Yeah, let's expand this. Okay.
00:49:53Substitute
00:49:552mx plus
00:50:06Well, this is right hand side. Put this left hand side. Okay.
00:50:17So,
00:50:21and left hand side is equal to
00:50:292mx plus four
00:50:43KN plus
00:50:51one.
00:51:04Two
00:51:09K plus N
00:51:15plus one.
00:51:20Okay.
00:51:51Okay.
00:52:11And we want this to be zero. Okay.
00:52:21Which is equal to two mx plus two KN
00:52:31plus K plus N plus one. Okay.
00:52:39And so we need to prove that Y exists, meaning K exists.
00:52:49And so we need to prove
00:52:58need
00:53:02there exists
00:53:07X and K.
00:53:12They're both integers such that
00:53:17we need this to be zero. Okay.
00:53:22For all m and integers
00:53:30mx plus two KN plus K plus N
00:53:37is equal to zero. Okay. So, this whole thing becomes one. Okay.
00:53:42Yeah.
00:53:47Okay.
00:54:05Maybe it's not that easy at all.
00:54:18X, K, K.
00:54:28Okay. So, again, we call it this one left prime,
00:54:36right prime. Okay. So, left prime is equal to
00:54:43mx plus K, two N plus one, plus N.
00:54:55Okay.
00:54:59Okay.
00:55:08You add number.
00:55:19Aye, aye, aye.
00:55:23Aye, aye, aye.
00:55:27Well, let's make some concrete examples. Okay. It's too abstract here.
00:55:33Okay.
00:55:39Truth.
00:55:45So, some even number and odd number.
00:55:51Okay. Some random even number and odd number.
00:55:57Let's say number six times X plus odd number.
00:56:03Let's say three times Y is equal to one.
00:56:09Okay.
00:56:21Okay.
00:56:27Okay.
00:56:35So, Y has to be an odd number.
00:56:52Okay.
00:57:04By the way, this conjecture could be wrong. Okay. So, yeah, if it's wrong,
00:57:08we'll disprove it and we'll define the conjecture. Okay. So, modify it.
00:57:13By the way, let me, you know what, let's take five minutes break. Okay.
00:57:18I'm going to check on my lunch. It's not as easy as I thought. Okay. So,
00:57:24sure.
00:57:28Okay. Yeah. Yeah. If you want to prove it or disprove it, go for it. Okay.
00:57:33Yeah. It's not as easy as I thought. Okay.
00:57:39All right. Five minutes. Thank you.
00:57:43Okay.
00:57:48Okay.
00:57:54Time check.
00:58:01It's been almost one hour. Let's ventilate the room.
00:58:18Okay.
00:58:48Okay.
00:59:18Okay.
00:59:48Okay.
01:00:18Okay.
01:00:49Okay.
01:00:52Okay.
01:01:16Well, I forgot to open the window, but we'll do that in the next break.
01:01:22Yeah.
01:01:27Okay.
01:01:30Let's find x and y.
01:01:36The y has to be odd number. Okay.
01:01:51Okay.
01:02:11Okay.
01:02:16Uh,
01:02:21Okay.
01:02:40Let's manipulate this. Okay.
01:02:45Uh, okay. Six times x is equal to
01:02:52one minus three y. Minus y, we'll call it z. Okay. So, one plus
01:03:01three z.
01:03:04Okay, and
01:03:07Okay.
01:03:13Okay.
01:03:17We just do trial and error.
01:03:21Okay. One plus three z, that's like one row three. Okay? One row three family.
01:03:30One row three.
01:03:32The kind of number, when it's divided by three, remainder is one. Okay? We have four.
01:03:41We have one, four. Um, just add three. Okay? So, seven, ten, thirteen, sixteen.
01:03:53We'll do this until we get a multiple of six. Okay? Yeah.
01:04:01Okay.
01:04:20Huh.
01:04:23Let me check my notes real quick.
01:04:43Well, this could be a counter example, actually.
01:04:47Will this ever be, like,
01:04:52Well, actually, this is,
01:05:11I don't think we'll ever become multiple of six. Okay?
01:05:16Oh.
01:05:20Wait a minute.
01:05:24Okay.
01:05:44So, six x has to be even number.
01:05:54Aye yai yai.
01:05:59It's kind of a deal fighting occasion.
01:06:12Yeah, maybe my conjecture was wrong. Okay.
01:06:16Well, that's fine. We can always modify our conjectures.
01:06:23Okay.
01:06:27Yeah.
01:06:30Okay, looks like we found a counter example to our conjecture. Okay, so.
01:06:34All right, that's fine.
01:06:47Okay.
01:06:58Then, the way we can modify our conjecture is this. Okay?
01:07:05Find this kind of circumstance where
01:07:10the linear combination of odd number and even number becomes one.
01:07:17Looks like it's not always the case. Okay, then when does that happen?
01:07:22Okay.
01:07:28Eh? Sure.
01:07:33There is something here. Number theory.
01:07:39Okay.
01:08:03Uh-huh.
01:08:10Okay.
01:08:21Ah, boy.
01:08:26The thing is this. I want to continue, but I'm just too hungry.
01:08:30Okay?
01:08:40Uh-huh.
01:08:58Sure.
01:09:09Okay.
01:09:12Okay.
01:09:36Out of curiosity, let's make one more example. Okay? How about
01:09:428x
01:09:45plus 5y
01:09:48is equal to 1. Does this exist?
01:10:13Well,
01:10:178x is equal to 1 plus 5z.
01:10:23I'm not sure whether we can do this, but for now.
01:10:29So z has to be odd number.
01:10:35So we have one row five.
01:10:43Oh, 16. Okay.
01:10:49z is equal to 3. Okay.
01:10:52y is equal to minus 3. Sure.
01:10:55Hm?
01:11:12Hm?
01:11:24Hm?
01:11:30Okay.
01:11:43Hm?
01:11:46Okay, so 6 is twice of 3. Okay, how about this?
01:11:5310x plus 5y is equal to 1. Does that exist?
01:12:02So 6 is twice of 3. Yeah, 10 is twice of 5. Okay?
01:12:20Yeah, that's not possible either. Okay.
01:12:26Okay, so we found two counterexamples. Good.
01:12:38So maybe this one does not work when this guy is multiple of this guy. Okay?
01:12:44That's a possibility. I don't know.
01:12:48We'll explore this later, but for now, let's take five minutes break.
01:12:54Let's take five minutes break. Okay, so I'm really hungry.
01:13:00Interesting problem, though. Okay? Yeah, yeah.
01:13:03So we found counterexample to this conjecture. Now, next step, we need to modify this conjecture to make it true.
01:13:10Okay? Kind of qualify, limit the scope. Okay? All right. Good. Making progress.
01:13:17Five minutes break. Okay? Thank you. Yeah.
01:13:20Okay.
01:13:29Yeah, it's kind of never-ending rabbit hole.
01:13:34Like Alice's Wonderland, I guess.
01:13:39All right. Five minutes. Thank you. Yep.
01:13:50Okay.
01:14:20Okay.
01:14:50Okay.
01:15:20Okay.
01:15:51Okay.
01:15:54Okay.
01:15:57Okay.
01:16:00Okay.
01:16:03Okay.
01:16:06Okay.
01:16:09Okay.
01:16:12Okay.
01:16:15Okay.
01:16:18Okay.
01:16:21Okay.
01:16:24Okay.
01:16:27Okay.
01:16:30Okay.
01:16:33Okay.
01:16:36Okay.
01:16:39Okay.
01:16:42Okay.
01:16:45Okay.
01:16:48Okay.
01:16:51Okay.
01:16:54Okay.
01:16:57Okay.
01:17:00Okay.
01:17:03Okay.
01:17:06Okay.
01:17:09So, we are dealing with dual-fountain equation.
01:17:15That's like Greek number theory tradition.
01:17:21Yeah, I encountered it when I was in middle school, high school, probably high school in South Korea. Okay?
01:17:30Okay.
01:17:34Yeah, we can check out Wikipedia
01:17:36for about two or five different occasions.
01:17:39So, in this case,
01:17:46okay, Y is,
01:17:50here it does not use it, but here it uses, okay?
01:17:53So, let's say X is two,
01:17:56and Y is minus two, okay?
01:18:04Yeah.
01:18:09Uh-huh.
01:18:11So, basically, eight times two, 16,
01:18:17plus five times,
01:18:21I'm sorry, minus three, okay?
01:18:23Minus three.
01:18:24Minus 15 is one, right?
01:18:27But also, there are other solutions
01:18:32to this different occasion, okay?
01:18:34So, like,
01:18:41maybe,
01:18:48seven comma, minus 11, okay?
01:18:52Yeah, like, two times five,
01:18:55I mean, two plus five, seven.
01:18:57Three plus eight, 11, okay?
01:18:59So, 56 minus 55, okay?
01:19:05Okay, infinite many solutions there.
01:19:06Okay, good, good.
01:19:11Okay.
01:19:14Yeah.
01:19:21Okay.
01:19:32Booyah.
01:19:38General solution,
01:19:43two and plus five,
01:19:47and,
01:19:52actually,
01:20:02five and plus two,
01:20:07and eight and minus three, okay?
01:20:12Minus three, okay?
01:20:18Well, minus eight and minus three, okay?
01:20:22So,
01:20:25uh-huh.
01:20:30Because eight times five and,
01:20:32and five times minus eight and,
01:20:34they cancel out, that's why, okay?
01:20:36Yeah.
01:20:37Okay.
01:20:42Okay.
01:20:50Let's wrap it up.
01:20:52Well, before we wrap it up,
01:20:54we'll check out the D.M. Fountain equation
01:20:56on Wikipedia, however.
01:20:57After that, let's wrap it up for this episode.
01:21:00My voice is hoarse, I'm hungry,
01:21:03my lunch is ready, okay?
01:21:05Cheers.
01:21:06Cheers.
01:21:11Yeah, let's go to Wikipedia.
01:21:13All right.
01:21:24Yep.
01:21:36Yeah, Bezos identity, yeah, linear D.M. Fountain equation.
01:21:39Okay.
01:21:43So we're on the right track, okay?
01:21:55Okay, so some results.
01:21:58Yeah.
01:22:04Okay, the solution, if and only if,
01:22:07sees a multiple of credits common divisor of A and B.
01:22:10Okay, good.
01:22:11Good to know, yeah.
01:22:15Huh.
01:22:28Okay.
01:22:40Okay.
01:22:46Nice.
01:22:48Good to know.
01:22:58Okay.
01:23:11Okay.
01:23:28Okay.
01:23:37Mm-hmm.
01:23:46I see, okay.
01:23:50So basically, there's no solution for this equation
01:23:55because three and six,
01:23:57the greatest common divisor is three, right?
01:24:05So one is not a multiple of three.
01:24:09Okay, so that's why it does not work here.
01:24:14Good.
01:24:15Nice.
01:24:16Good to know, okay?
01:24:19Okay.
01:24:21Okay.
01:24:25It's good that we learned that
01:24:27because it gives us some guidance, okay, in this journey
01:24:32so that we do not have to, like,
01:24:37reprove what they already proved, right?
01:24:40Yeah, so it's good.
01:24:42Saving us a lot of time here.
01:24:44Yeah, good, good.
01:24:46Okay.
01:24:47Nice.
01:24:51Okay.
01:24:52Then what would be our next step?
01:24:57We're gonna modify our concept, okay?
01:24:59So based on that known theorem, okay?
01:25:03Yeah.
01:25:04It's really good to know.
01:25:05It helps.
01:25:06Nice.
01:25:07Okay.
01:25:08Very cool.
01:25:09Mm-hmm.
01:25:11Okay.
01:25:20Okay.
01:25:26So let me slip on this, okay?
01:25:31Yeah, given that theorem we just looked at in the Wikipedia,
01:25:37yeah, I'm gonna modify the concept chart,
01:25:39and then we'll go from there.
01:25:44Okay?
01:25:45All right.
01:25:46See you tonight, okay?
01:25:47Yeah.
01:25:48Yeah.
01:25:49Just to study what other people discovered.
01:25:53Yeah, nice.
01:25:54Okay.
01:25:55Yeah, see you.
01:25:56Yeah.
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