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Huhnkie Lee
  • 2 days ago
continuing journey in number theory
Transcript
00:00:00Hello friends, welcome to Humanology episode 2259, happy Tuesday evening, yeah let me get
00:00:29started.
00:00:30Today I applied to some jobs, like there have been some big corporations in Alaska that
00:00:50I wanted to, I was, I have been interested in the past, so now that I'm unemployed, yeah
00:00:57actually I'm applying to those big corporations in Alaska, yeah so it's all good, thank you.
00:01:04Okay, let's do stretching, yeah I woke up and went outside and did some running, nice.
00:01:24Okay.
00:01:27Ah.
00:01:31Ah.
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00:02:08Oh
00:02:26So like pasta dry pasta
00:02:29last very long because they become really dry also
00:02:33grains like beans
00:02:35or other grains like rice
00:02:41Not standard to less bellow. Yeah
00:02:47Dried carbohydrates. Okay
00:02:52Because there was this kind of like a one-third box of
00:02:59container pasta that was left
00:03:01from years ago. I cooked it. It's still good
00:03:05Oh
00:03:08It's like a word
00:03:25Oh
00:03:32Oh
00:03:34Hmm.
00:04:01Yeah.
00:04:34Oh, face exercise.
00:05:04Okay.
00:05:11Nope.
00:05:12Nope.
00:05:13Nope.
00:05:15Nope.
00:05:19Nope.
00:05:21Nope.
00:06:23Now, let's do one leg calf lifting.
00:06:39Okay.
00:07:09One more time.
00:07:13One more time.
00:07:43Okay.
00:07:43Good, good, good.
00:07:44Ooh.
00:07:45Five minutes, Brad.
00:07:53Okay.
00:07:54One more time.
00:08:02One more time.
00:08:02One more time.
00:08:03One more time.
00:08:04One more time.
00:08:05One more time.
00:08:05One more time.
00:08:06One more time.
00:08:06One more time.
00:08:07One more time.
00:08:07One more time.
00:08:08One more time.
00:08:08One more time.
00:08:09One more time.
00:08:09One more time.
00:08:10One more time.
00:08:11One more time.
00:08:12One more time.
00:08:13One more time.
00:08:13One more time.
00:08:14One more time.
00:08:15One more time.
00:14:16Okay, so sorry about the suboptimal hair, but I'll take a bath at some point, okay?
00:14:28So, okay.
00:14:30Okay.
00:14:32Yeah.
00:14:34This is, I think, I take a bath like once a week, okay, because I'm a minimalist.
00:14:40Okay, now, okay, now, what kind of numbers are there.
00:14:50Okay, okay, right?
00:14:58Okay, okay, right?
00:15:02Okay
00:15:05Right
00:15:09We have this original primary basic coefficient right minus 2 and 1
00:15:15basically when we do
00:15:173x plus 7y is equal to 2 we basically multiply by 2 in those alpha and beta so 2 minus 2 becomes minus 4
00:15:271 becomes 2
00:15:29Okay
00:15:31Yeah
00:15:34When we do multiply by 3
00:15:38Minus 2 becomes minus 6 1 becomes 3. Yeah multiply by 3. Okay, that make perfect sense
00:15:46When we do multiply by 4
00:15:502 times 4 is 8
00:15:53But 8 rho 7 is 1. Okay, that's why we have minus 7 l minus 1. Okay
00:16:01Now
00:16:021 times 4 is 4 but 4 rho 3 is 1. That's why we have plus 1 here. Okay, so
00:16:09It makes perfect sense
00:16:16Yeah
00:16:18So
00:16:23Now
00:16:27This benzo coefficients
00:16:29When l is when 3x plus 7y is equal to 2
00:16:36This pair of x and y
00:16:39Sometimes they are like even number even number like minus 1 and 2
00:16:43Sometimes they are odd number and odd number like
00:16:51Minus 11 and 5
00:16:54Because 3 and 7 they are odd number odd number right to make the
00:16:59Even number out of two numbers addition
00:17:02They have to be either
00:17:04Add add add add plus add or even plus even
00:17:07In order to make even number two
00:17:09Okay, so that makes sense
00:17:12Okay
00:17:14Mm-hmm
00:17:32Okay
00:17:36Interesting
00:17:39Now
00:17:44You're gonna
00:17:46I need to draw that matrix again. Okay, for number seven. Let me see if we already have it somehow
00:18:02I look for it
00:18:25We had we do have it
00:18:28Right here
00:18:29Okay, let's take five minutes break. Okay, so and then
00:18:37Look at that
00:18:38Remainder matrix together again
00:18:41Okay, yeah, I did this is my short pants. Okay, it's not on the wire
00:18:49Okay
00:18:51Uh
00:18:57Five minutes break. Thank you
00:19:05Okay
00:19:07You
00:19:21You
00:19:23You
00:19:25You
00:19:27You
00:19:29You
00:19:31You
00:19:42You
00:19:44You
00:19:46You
00:19:47You
00:19:50You
00:19:50You
00:19:52You
00:22:31Okay.
00:22:32Let's switch to whiteboard.
00:22:34And...
00:22:35Yeah.
00:22:42Yeah.
00:22:43Now we are looking at different rows of this remainder matrix.
00:22:51I mean, we can call it Dezu matrix, but let's not give him too much credit, okay?
00:22:58He didn't discover this matrix.
00:22:59We did.
00:23:00Then would I name it like honky matrix?
00:23:05No.
00:23:06We discovered many matrix and...
00:23:08We just call it the remainder matrix, okay?
00:23:13Okay.
00:23:14Not human knowledge matrix either, okay?
00:23:16We discovered so many matrix so far.
00:23:20Now, in the past, previously, we only looked at the last row because 3x plus... well, ax plus 7y is equal to zero, right?
00:23:283x plus... well, ax plus 7y is equal to zero, right?
00:23:40Yeah.
00:23:41Yeah.
00:23:43Yeah.
00:23:45Yeah.
00:23:46This is small whiteboard.
00:23:48That's why I cannot quite put the pen holder in this one.
00:23:54So, we have ax plus 7y is equal to 1, that's the last row, okay?
00:24:22Second to the last row, that's like ax plus 7y is equal to 2, okay?
00:24:32Now, let's look at alpha there in the other whiteboard, okay?
00:24:39Do they match?
00:24:42Uh... we have... we have 3... we have 3x plus 7y is equal to... we look at 2... yeah, 4 matches.
00:24:59And, um... 3... yeah, 6, that matches.
00:25:054... 1.
00:25:08It does match.
00:25:09As expected.
00:25:11Good.
00:25:12I'm glad it does.
00:25:14Good, good.
00:25:16Good.
00:25:17Good.
00:25:18Good.
00:25:19Good.
00:25:20Good.
00:25:21Good.
00:25:22Good.
00:25:23Good.
00:25:24Good.
00:25:25Good.
00:25:26Good.
00:25:27It's something we didn't like quite extensively study before, that's why I was a little bit...
00:25:33not concerned, but just a little bit concerned actually.
00:25:44Okay, just a little bit.
00:25:45But it does match.
00:25:46Good.
00:25:47I'm glad.
00:25:48Yeah.
00:25:49Maybe we have done it before in the past, but I don't think we did quite like we did yesterday.
00:25:58Okay, so we are just basically studying in this matrix in a more holistic point of view.
00:26:05Okay, like how to come back from one row to another row.
00:26:09Yeah, now we know how to do that.
00:26:11Okay, so basically multiply 2, multiply 3, 4, 5, 7, and then row 7.
00:26:24Okay?
00:26:252, for example.
00:26:262 times 2, 3, 4, 5, 6.
00:26:312 times 6.
00:26:3416.
00:26:3516 row 7.
00:26:375.
00:26:38Okay?
00:26:40Same thing.
00:26:425 times 6.
00:26:4330.
00:26:4430 row 7.
00:26:452.
00:26:46Okay?
00:26:47So it works beautifully.
00:26:48Okay?
00:26:49Nice.
00:26:50Yeah.
00:26:51And it's not a coincidence.
00:26:54We kind of know why it works.
00:26:57Okay?
00:26:58Not in detail yet though.
00:27:00There's a lot going on here.
00:27:02Okay?
00:27:03So we're just having better understanding than before.
00:27:06Okay?
00:27:07So, good.
00:27:08Uh-huh.
00:27:09Okay.
00:27:10Uh-huh.
00:27:11Okay.
00:27:12Okay.
00:27:13Okay.
00:27:14Okay.
00:27:15Okay.
00:27:16Okay.
00:27:17Okay.
00:27:18Now, interesting thing is this.
00:27:19Okay?
00:27:20Uh-huh.
00:27:21Okay.
00:27:22Uh-huh.
00:27:23Now, interesting thing is this.
00:27:27Okay?
00:27:28Uh-huh.
00:27:29Uh-huh.
00:27:30Uh-huh.
00:27:31We have, let's say, uh-huh.
00:27:35So we have, let's say,
00:27:58let me discard this.
00:28:01Bonk.
00:28:08And watch my hands.
00:28:11We have 3x plus 7y is equal to 2.
00:28:26We have 3x plus 7y is equal to 2.
00:28:41General solution,
00:28:43x is equal to minus 7k,
00:28:48minus 4,
00:28:50y is equal to 3k plus 2.
00:28:54Now, but there's some odd pairing of this x and y.
00:29:03Let's call it secondary best coefficients, okay?
00:29:06x and y.
00:29:09If k is 1,
00:29:11x is equal to 2.
00:29:15Okay, we have minus 11.
00:29:17Minus 11 and 5, so minus 3 times 11 plus 7 times 5 is equal to 2, okay?
00:29:47All right, and 3 and 7, they are co-primes, 5 and 11, they are co-primes too, okay?
00:30:1411 and 7, they are co-primes, 3 and 5, they are co-primes too, 5, 7, 3, 11, they are all
00:30:29prime numbers, so that's why they are pair-wise co-primes, okay?
00:30:33All right, serious.
00:30:40And they are all odd numbers, they are not even numbers, none of them are multiples of 2, okay?
00:30:49But still, when we add these two terms, it still adds up to 2, which is an even number.
00:30:58We have basically odd integer minus odd integer, that's why we have an even number, okay?
00:31:04So, and in this case, it works because kind of like Bezos theorem, whatever theorem, it says this
00:31:18deophantification works when the right-hand side here is multiple of gamma GCD of the two numbers.
00:31:27Here, the gamma GCD is 1, okay?
00:31:31And now 2 is 1 times 2, okay?
00:31:34So it does not violate that Bezos rule, okay?
00:31:38Yeah, sure.
00:31:40But strange thing, none of them are even numbers, okay?
00:31:44So, yeah, that is something we did not consider in the past.
00:31:49We kind of realized this case yesterday.
00:31:54All right, yeah, cheers.
00:31:59Yeah, which is good to know.
00:32:09So, Ax plus By is equal to 2.
00:32:13That does not necessarily mean the greatest common divisor of x and y is 2.
00:32:20No, they could be 1.
00:32:22Like in this case.
00:32:23You know, 5 and 11, deophantification is 1, not 2, okay?
00:32:27It's just 2 is a multiple of 1.
00:32:371 times 2, okay?
00:32:38Yeah.
00:32:39Okay.
00:32:40Good to know.
00:32:44We have to be careful about this, okay?
00:32:47So, okay.
00:32:50Okay.
00:32:57So, we have this k, k is an integer, and meaning it can be odd number.
00:33:26Equally even number.
00:33:29If k is odd number, then both x and y are odd numbers.
00:33:38If k is an even number, then both x and y are even numbers.
00:33:43Okay?
00:33:45And in this schema, when k is an even number,
00:33:56then those are the numbers where x and y, when we have x and y, we have created this common divisor of 2.
00:34:14Not 4 or 6, okay?
00:34:15Why?
00:34:16In this form, okay.
00:34:17Let's say k is 2 times l, okay?
00:34:19Even number.
00:34:20Even number.
00:34:21Then we have x is equal to minus 2.
00:34:25We have x is equal to minus 2.
00:34:26We are putting 2 l instead.
00:34:27Okay.
00:34:28Okay.
00:34:29Okay.
00:34:30So, 7 l minus 2.
00:34:31Okay.
00:34:32So, we have x is equal to minus 2.
00:34:34We are putting 2 l instead.
00:34:35Okay.
00:34:36Okay.
00:34:37So, 7 l minus 2.
00:34:38Okay.
00:34:39So, we have x is equal to minus 2.
00:34:40Okay.
00:34:41So, we have x is equal to minus 2.
00:34:43Okay.
00:34:44Okay.
00:34:45Okay.
00:34:46So, we have x is equal to minus 2.
00:34:49So, we have x minus 2.
00:34:52Okay.
00:34:53Okay.
00:34:54So, 7 l minus 2.
00:34:57Okay.
00:34:58Y is equal to, 2, 3 l plus 1.
00:35:06Okay.
00:35:08So, we have x.
00:35:10Now
00:35:16The 3L minus 2
00:35:307L minus 2
00:35:35and 3L plus 1
00:35:39Can
00:35:56Are there always
00:35:58co-primes?
00:36:07Are there
00:36:091L?
00:36:101L.
00:36:111L
00:36:131L
00:36:15That's supposed to be
00:36:16OK
00:36:17Yeah
00:36:18Yeah
00:36:20Just when, even when k is an even number
00:36:24Okay, three, x and y, okay, x and y creates common divisors, too, only when k is an even number, okay?
00:36:54And so they cannot have any other greatest common divisor, x and y, other than two, and that happens only when k is an even number.
00:37:10So k is 2l, an even number, okay? And then we have this, okay?
00:37:16Okay, and then means 7l minus 2 and 3l plus 1, they have to co-prime to each other, okay?
00:37:25And let's make some examples, let's see if that's the case. Of course, we need another y-port, okay? That's fine.
00:37:32I'll play a y-port eraser, then I can erase, okay? Five minutes, Greg, okay? Thank you. How interesting, how interesting.
00:37:41Okay.
00:37:48Okay.
00:37:50Okay.
00:37:55Okay.
00:38:00Okay.
00:40:335-4-2-8.
00:40:35That's okay.
00:40:35Let's take a brief read for mathematics.
00:41:05Mosquitoes, they are attacked to running cars.
00:41:08Do you know why?
00:41:08Because mosquitoes, they are attracted to animals, basically, like heat and carbon dioxide, okay?
00:41:14Like, if you smell something, we can kind of track down the origin, where does the smell come from, like it could be some animal fizzes, whatever.
00:41:27So, mosquitoes, yeah, they detect a chemical gradient, just like humans do when we detect smell, okay?
00:41:39So, yeah, cars, just like, they're like animals, okay?
00:41:43So, they are warm, running cars, and then emit carbon dioxide, okay?
00:41:50So, cars are like animals, that's why mosquitoes are attracted to cars, okay?
00:41:53All right.
00:41:54All right.
00:41:55All right.
00:41:56All right.
00:41:57Let's erase this.
00:41:59And let me copy down what we have over there.
00:42:22And let me copy down what we have over there.
00:42:52Okay, it's good to 2L and case even
00:43:03Okay
00:43:21L is a number, okay
00:43:31So we can modify this. Let's say L is equal to
00:43:36M plus 1, okay
00:43:42M plus 1
00:43:48Okay
00:43:55Maybe not. Well, we'll consider that later. Okay
00:44:12Okay
00:44:33I'm going to start drinking a little bit
00:44:38Okay, yes, right
00:44:45Oh yeah, cheers
00:44:47Okay, this is whiskey
00:44:49Let me have some tongs
00:44:55Let me have some tongs
00:44:59Strong, strong
00:45:01All right
00:45:06Okay
00:45:17Okay
00:45:19So
00:45:283
00:45:30I mean 7L minus 2
00:45:32And 3L plus 1
00:45:35They are supposed to be co-primes
00:45:36To each other
00:45:37No matter what L is
00:45:41Which is quite surprising. Okay, so
00:45:42And let's test them out
00:45:49If L is 0, yeah
00:45:51Signs, ignore the signs. Okay, yeah
00:45:542 and 1, yeah, they are co-primes
00:45:56When L is 1
00:46:13Let's throw table
00:46:13Sure
00:46:14Sure
00:46:14Sure
00:46:14Sure
00:46:14Sure
00:46:14Sure
00:46:14L
00:46:190
00:46:197L minus 2
00:46:353L plus 1
00:46:41Okay
00:46:42Okay
00:47:02Okay
00:47:12They are not co-primes, oops, okay.
00:47:42Okay, so we found some counter example, okay.
00:48:08Then let's see what's going on here.
00:48:12That's good enough, okay, we don't need more examples.
00:48:26So, k is 8, l is 4, x is equal to, I memorized the multiplication table in Korean.
00:48:49Okay, so I have some internal translation from Korean to English, okay, multiplication table.
00:48:56So, 7 times 8, 56, okay, so, minus 60, y is equal to, when k is 8, okay, yeah.
00:49:14So, 3 times 8, 24, 26, okay.
00:49:30Okay, now 3 times x, plus 7 times y.
00:49:40Okay, okay, okay.
00:49:47What's going on?
00:49:54Minus 3 times 60, plus 7 times 10 is 6, is equal to 2.
00:50:02Minus, minus 3 times 60, plus 7 times 10 is 6, is equal to 2.
00:50:22So, 60 and 26, yes.
00:50:24Yes, their most common device is 2.
00:50:32Okay.
00:50:33Yeah, yeah.
00:50:35Cheers.
00:50:37Thanks.
00:51:48I mean, according to this, x is minus 2 times this, so minus 52.
00:52:13Why is it 2 times of this?
00:52:22I'm missing something.
00:52:23It's not quite adding up.
00:52:24I made a mistake somewhere, because according to one calculation, x is minus 52.
00:52:41Another calculation, x is minus 60.
00:52:44And this should be correct and this should be incorrect.
00:52:48And I am not quite getting it.
00:52:52I got it.
00:53:02This is positive 2.
00:53:06So, we're taking up to minus 2, okay?
00:53:09So, okay.
00:53:12Okay.
00:53:17Then.
00:53:27Maybe they are cool problems after all, huh?
00:53:29Yep, my bad.
00:53:37So, 7L plus 2.
00:53:493L plus 1.
00:53:50So, yeah.
00:54:02So, yeah.
00:54:03Okay.
00:54:03So, yeah.
00:54:34There are co-primes, okay, okay.
00:54:38Always.
00:54:44Can we prove it?
00:54:45We kind of did, okay?
00:54:51That's like a basal, it's a
00:54:55corollary result, implication from basal theorem, okay, well, yeah.
00:55:04We kind of proved it, but I'm kind of wondering why it works so well.
00:55:197L plus 2 and 3L plus 1, they are always, always co-primes, okay, which is really cool,
00:55:25okay.
00:55:26Yeah, I mean, that's not obvious at all, you know.
00:55:31Yeah.
00:55:32Oof, interesting.
00:55:38But can we prove this in some other ways?
00:55:45Well, 7L plus 2 is co-prime with 3L plus 1, that is not obvious at all.
00:55:52Yeah, I mean, we can do subtraction, right?
00:55:59Like, co-prime audition, subtraction theorem, right?
00:56:06And, which is powerful because it's bi-directional.
00:56:10Yeah.
00:56:11Yeah.
00:56:12I mean, we can do subtraction, right?
00:56:14Like, co-prime audition, subtraction theorem, right?
00:56:20And, which is powerful because it's bi-directional.
00:56:35Subtract.
00:56:45Okay.
00:56:50Yeah.
00:56:51Subtract.
00:56:52L three times.
00:57:05Well, it goes like a...
00:57:20Okay.
00:57:21Yeah.
00:57:22There we go.
00:57:23Yes.
00:57:24It's another way to prove it, okay?
00:57:25Sure, sure, sure.
00:57:26So, we can prove this two ways, okay?
00:57:27Good, good.
00:57:28Cheers.
00:57:29Yeah.
00:57:30This is interesting.
00:57:31Yeah.
00:57:32Of course, we can go from here.
00:57:35Yeah.
00:57:36Yeah.
00:57:37Yeah.
00:57:38Yeah.
00:57:39Yeah.
00:57:40Yeah.
00:57:41Yeah.
00:57:42Yeah.
00:57:43Yeah.
00:57:44Yeah.
00:57:45Yeah.
00:57:46Yeah.
00:57:47Yeah.
00:57:48Yeah.
00:57:49Yeah.
00:57:50Of course, we can go from here to...
00:57:51In terms of subtracting it, multiply by two and then subtract, right?
00:57:56Yeah.
00:57:57So, like, it would become like...
00:57:58It will become like...
00:58:16Stuff like that. Okay, okay, okay. Interesting.
00:58:25No, time check.
00:58:28It's been one hour. Okay, yeah. Let's see if I'm is very good.
00:58:36Interesting. Yeah, learning more about just numbers, number theory world, integer world.
00:58:49Sure. Yeah, slow progress, but that's fine. We have time.
00:58:55Yeah, we are making progress in our learning.
00:59:01That's good enough. Okay? Yeah.
00:59:03Five minutes. Thank you.
01:01:30Okay.
01:01:39Uh-huh.
01:01:40So, let's now look at the big picture, okay?
01:01:49Because, again, this is like a forest.
01:01:51We are like hiking.
01:01:53We are like gold miners exploring the forest to find a source of gold.
01:02:00We are kind of like learning the terrain, kind of mapping the terrain, like landmarks, whatever.
01:02:08Okay, where is the creek bed, where is, like, mountain range?
01:02:14We are kind of mapping the terrain, and then in the creek bed, yeah, you see there are some golds.
01:02:21That's called something like placer, placer gold, okay?
01:02:25Because if there are some gold in the river bed, that means when we go upstream, they're possibly a source of gold.
01:02:34This gold is something up the mountain, upstream, you know?
01:02:42So, that's how they find gold mines, okay?
01:02:45So, I know about this, but I read about it.
01:02:49Alaskan gold miners stories, okay?
01:02:51This is fascinating, okay?
01:02:52Now, now, let's look at the big picture, okay?
01:02:56Bird's eye view, okay?
01:02:58Maybe some drone these days, okay?
01:03:01Yeah, drones, like, you can do this camera, right?
01:03:05Yeah, so, we can, like, see what drone capture in the camera, motion picture, okay?
01:03:11So, and, this is really cool stuff, okay?
01:03:15So, like, a while ago, a long time ago,
01:03:22or several months ago, one possibility that we came up with is this, okay?
01:03:28So, if we know basal coefficient, maybe we can calculate the greatest common divisor from that.
01:03:34And we have generally formula, not just for, like, primary basal coefficient, like, ax plus byx goes to one.
01:03:45No, we have basal coefficient formula for everything.
01:03:48Ax plus by is equal to two, three, four.
01:03:54We have formula for everything, okay?
01:03:59The remainder matrix element formula.
01:04:03The ax plus by is equal to one.
01:04:06That's just last row.
01:04:08But we have formula for everything, the remainder matrix, okay?
01:04:14So, yeah, cheers.
01:04:17What we found is something very profound.
01:04:21It rhymes on.
01:04:23Yeah, cheers.
01:04:24Yeah, yeah.
01:04:34Hmm?
01:04:35Hmm?
01:04:35Hmm?
01:04:35Hmm?
01:04:35Hmm?
01:04:38Hmm, but we also noticed a case where...
01:04:47Let's write it one more time, okay?
01:04:59Okay?
01:05:00Here, ax plus by is equal to two, but two is not a greatest common divisor of 11 and 5, or 3 and 7.
01:05:09And two is not a greatest common divisor of any combination of these numbers, okay?
01:05:13Okay?
01:05:14So, we need to keep these cases in mind, okay?
01:05:17Yeah.
01:05:21But here we distinguish that.
01:05:23This happens when k is an odd number in this particular case, okay?
01:05:32When k is even number, then yeah, the greatest common divisor between the two numbers are always true.
01:05:41Why?
01:05:42Because we prove that these remnants, okay?
01:05:43Because we prove that these remnants...
01:05:45Remnants...
01:05:46Remnants...
01:05:47What is remnants?
01:05:48It's kind of a reminder in multiplication context.
01:05:51Let's call it remnants, okay?
01:05:52Let's call it remnants, okay?
01:05:53Yeah, these two remnants, they are co-parent to each other, and we just proved it.
01:05:56Okay?
01:05:57Yeah.
01:05:58We are making progress here, okay?
01:06:00Cheers.
01:06:01We are making progress here, okay?
01:06:02Cheers.
01:06:03We are making progress here, okay?
01:06:04Yeah.
01:06:05Cheers.
01:06:06Okay.
01:06:08I'll grab some video.
01:06:14Okay?
01:06:15We are not out of the woods yet...
01:06:26Well, I'm not.
01:06:27I'm not.
01:06:28where i'm not maybe you can discover constant time uh formula for great common divisor before
01:06:35i do go for it okay yeah go for it oh yeah yeah sure and feel free to publish before i do okay yeah
01:06:46no problem okay i just give it some due credit uh to humanology and honkily okay so okay okay
01:06:57in a full note okay yeah cheers
01:07:03okay yeah
01:07:07uh but for me i'm not out of those yet but that's fine
01:07:12this is supposed to be a difficult problem that nobody else solved for more than 200
01:07:192 000 years all around the world it's supposed to be a difficult problem okay
01:07:28so uh you're kind of making slow progress but that's fine
01:07:35you got time
01:07:35uh
01:07:43doesn't know rush
01:07:44not me
01:07:49yeah
01:07:54we're enjoying the journey okay yeah
01:08:07the way toward that goal okay yeah
01:08:14this we already discovered basic coefficient formula for all cases
01:08:20the entire remainder matrix every single element yeah we have formula
01:08:25because it's generally formula that applies to all those
01:08:28element remainder matrix okay and from there we also found constant
01:08:35that's just constant time formula okay yeah remainder matrix formula okay
01:08:41element formula and then we also have based on that formula
01:08:45co-primality formula which is also constant time
01:08:49we already discovered a lot of good things already
01:08:52okay and now we want to raise the bar
01:08:56and we want to find constant time formula for
01:09:00greatest common divisor and I do
01:09:02believe that's possible that it exists okay
01:09:08if it does exist we'll find it
01:09:13okay yeah cheers
01:09:16yeah
01:09:17mm-hmm
01:09:23mm-hmm
01:09:27mm-hmm
01:09:29mm-hmm
01:09:30mm-hmm
01:09:31mm-hmm
01:09:41mm-hmm
01:09:43I feel that we did enough mathematics okay and I need some vocalist okay so it's been more than one hour okay so uh let's uh digitize this and let's go to after that let's go to this time live okay
01:09:44mm-hmm
01:09:45mm-hmm
01:09:46mm-hmm
01:09:47mm-hmm
01:09:48mm-hmm
01:09:49mm-hmm
01:09:50mm-hmm
01:09:51mm-hmm
01:09:52mm-hmm
01:09:53mm-hmm
01:09:54mm-hmm
01:09:55mm-hmm
01:09:56mm-hmm
01:09:58mm-hmm