title

Pi hiding in prime regularities

description

A story of pi, primes, complex numbers, and how number theory braids them together.
Mathologer on why 4k + 1primes break down as sums of squares: https://youtu.be/DjI1NICfjOk
Help fund future projects: https://www.patreon.com/3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: http://3b1b.co/leibniz-thanks
Home page: https://www.3blue1brown.com/
For those of you curious about the finer details, here's a writeup from the viewer Daniel Flores justifying the final approximation: https://www.overleaf.com/read/wdzkfjbkwzyf
The fact that only primes that are one above a multiple of four can be expressed as the sum of two squares is known as "Fermat's theorem on sums of two squares": https://goo.gl/EdhaN2
Music by Vince Rubinetti:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
Timestamps
0:00 - Introduction
1:39 - Counting lattice points
5:47 - Gaussian integers
10:30 - The lattice point recipe
17:50 - Counting on one ring
20:14 - Exploiting prime regularity
25:19 - Combining the rings
28:36 - Branches of number theory
------------------
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detail

{'title': 'Pi hiding in prime regularities', 'heatmap': [{'end': 1184.057, 'start': 1152.715, 'weight': 1}], 'summary': 'Delves into the interconnection between prime numbers, complex numbers, and pi, revealing a formula for pi as an alternating infinite sum. it discusses lattice points, gaussian integers, factorization, and the counting of lattice points on circles, highlighting their ties to the distribution of primes and their connection to pi.', 'chapters': [{'end': 143.848, 'segs': [{'end': 36.27, 'src': 'embed', 'start': 4.423, 'weight': 2, 'content': [{'end': 6.763, 'text': "This is a video I've been excited to make for a while now.", 'start': 4.423, 'duration': 2.34}, {'end': 13.325, 'text': 'The story here braids together prime numbers, complex numbers, and pi in a very pleasing trio.', 'start': 7.344, 'duration': 5.981}, {'end': 22.247, 'text': 'Quite often in modern math, especially that which flirts with the Riemann zeta function, these three seemingly unrelated objects show up in unison,', 'start': 14.025, 'duration': 8.222}, {'end': 28.708, 'text': "and I want to give you a little peek at one instance where this happens, one of the few that doesn't require too heavy a technical background.", 'start': 22.247, 'duration': 6.461}, {'end': 36.27, 'text': "That's not to say that this is easy, in fact this is probably one of the most intricate videos I've ever done, but the culmination is worth it.", 'start': 29.548, 'duration': 6.722}], 'summary': 'The video explores the connection between prime numbers, complex numbers, and pi in the realm of modern math, particularly the riemann zeta function.', 'duration': 31.847, 'max_score': 4.423, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY4423.jpg'}, {'end': 70.167, 'src': 'embed', 'start': 42.572, 'weight': 3, 'content': [{'end': 47.453, 'text': "This formula is actually written on the mug that I'm drinking coffee from right now as I write this,", 'start': 42.572, 'duration': 4.881}, {'end': 56.176, 'text': 'and a fun but almost certainly apocryphal story is that the beauty of this formula is what inspired Leibniz to quit being a lawyer and instead pursue math.', 'start': 47.453, 'duration': 8.723}, {'end': 63.743, 'text': "Now whenever you see pi show up in math, there's always going to be a circle hiding somewhere, sometimes very sneakily.", 'start': 57.219, 'duration': 6.524}, {'end': 70.167, 'text': 'So the goal here is not just to discover this sum, but to really understand the circle hiding behind it.', 'start': 64.343, 'duration': 5.824}], 'summary': 'The beauty of the pi formula inspired leibniz to pursue math, always a circle hiding behind it.', 'duration': 27.595, 'max_score': 42.572, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY42572.jpg'}, {'end': 116.868, 'src': 'embed', 'start': 85.275, 'weight': 0, 'content': [{'end': 88.996, 'text': 'but not really getting a sense for why or for where the hidden circle is.', 'start': 85.275, 'duration': 3.721}, {'end': 91.997, 'text': 'On the path that you and I will take, though,', 'start': 89.916, 'duration': 2.081}, {'end': 102.621, 'text': "what you'll see is that the fundamental truth behind this sum and the circle that it hides is a certain regularity in the way that prime numbers behave when you put them inside the complex numbers.", 'start': 91.997, 'duration': 10.624}, {'end': 110.585, 'text': 'To start the story, imagine yourself with nothing more than a pencil, some paper, and a desire to find a formula for computing pi.', 'start': 103.681, 'duration': 6.904}, {'end': 116.868, 'text': 'There are countless ways you could approach this, but as a broad outline for the plotline here,', 'start': 111.545, 'duration': 5.323}], 'summary': 'Discover the fundamental truth behind the hidden circle and the sum, exploring prime numbers inside complex numbers to find a formula for computing pi.', 'duration': 31.593, 'max_score': 85.275, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY85275.jpg'}], 'start': 4.423, 'title': 'The riemann zeta function and the formula for pi', 'summary': 'Explores the interconnection between prime numbers, complex numbers, and pi, culminating in a formula for pi as a certain alternating infinite sum, revealing the hidden circle and the connection to the way prime numbers behave in complex numbers.', 'chapters': [{'end': 143.848, 'start': 4.423, 'title': 'The riemann zeta function and the formula for pi', 'summary': 'Explores the interconnection between prime numbers, complex numbers, and pi, culminating in a formula for pi as a certain alternating infinite sum, revealing the hidden circle and the connection to the way prime numbers behave in complex numbers.', 'duration': 139.425, 'highlights': ['The fundamental truth behind the formula for pi and the hidden circle is a certain regularity in the way that prime numbers behave when placed inside the complex numbers.', 'The story braids together prime numbers, complex numbers, and pi, leading to a formula for pi as an alternating infinite sum, with the culmination being worth the intricacy of the video.', 'The formula for pi is derived from a certain regularity in the behavior of prime numbers when placed inside the complex numbers, providing a deeper understanding of the circle hidden behind it.', 'The formula for pi is written on the mug, with a story suggesting that its beauty inspired Leibniz to pursue math instead of law, signifying the impact and elegance of the formula.']}], 'duration': 139.425, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY4423.jpg', 'highlights': ['The formula for pi is derived from a certain regularity in the behavior of prime numbers when placed inside the complex numbers, providing a deeper understanding of the circle hidden behind it.', 'The fundamental truth behind the formula for pi and the hidden circle is a certain regularity in the way that prime numbers behave when placed inside the complex numbers.', 'The story braids together prime numbers, complex numbers, and pi, leading to a formula for pi as an alternating infinite sum, with the culmination being worth the intricacy of the video.', 'The formula for pi is written on the mug, with a story suggesting that its beauty inspired Leibniz to pursue math instead of law, signifying the impact and elegance of the formula.']}, {'end': 453.436, 'segs': [{'end': 294.902, 'src': 'embed', 'start': 267.343, 'weight': 0, 'content': [{'end': 275.327, 'text': 'And what we want is a systematic way to count how many lattice points are on a given one of these rings, a given distance from the origin,', 'start': 267.343, 'duration': 7.984}, {'end': 276.488, 'text': 'and then to tally them all up.', 'start': 275.327, 'duration': 1.161}, {'end': 285.099, 'text': "And if you pause and try this for a moment, what you'll find is that the pattern seems really chaotic, just very hard to find order under here.", 'start': 277.677, 'duration': 7.422}, {'end': 289.24, 'text': "And that's a good sign that some very interesting math is about to come into play.", 'start': 285.659, 'duration': 3.581}, {'end': 294.902, 'text': "In fact, as you'll see, this pattern is rooted in the distribution of primes.", 'start': 290.141, 'duration': 4.761}], 'summary': 'Count lattice points on rings, rooted in prime distribution.', 'duration': 27.559, 'max_score': 267.343, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY267343.jpg'}, {'end': 370.58, 'src': 'embed', 'start': 338.67, 'weight': 2, 'content': [{'end': 344.277, 'text': 'And that corresponds to the fact that you cannot find two integers whose squares add up to 11.', 'start': 338.67, 'duration': 5.607}, {'end': 344.557, 'text': 'Try it.', 'start': 344.277, 'duration': 0.28}, {'end': 353.061, 'text': 'Now many times in math, when you see a question that has to do with the 2D plane,', 'start': 348.536, 'duration': 4.525}, {'end': 359.287, 'text': 'it can be surprisingly fruitful to just ask what it looks like when you think of this plane as the set of all complex numbers.', 'start': 353.061, 'duration': 6.226}, {'end': 370.58, 'text': 'So, instead of thinking of this lattice point here as the pair of integer coordinates, instead think of it as the single complex number.', 'start': 360.308, 'duration': 10.272}], 'summary': "In math, no two integers' squares add up to 11. consider 2d plane as complex numbers.", 'duration': 31.91, 'max_score': 338.67, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY338670.jpg'}, {'end': 408.336, 'src': 'embed', 'start': 382.709, 'weight': 1, 'content': [{'end': 388.173, 'text': "It's what you get by reflecting over the real axis, replacing i with negative i.", 'start': 382.709, 'duration': 5.464}, {'end': 392.456, 'text': "And this might seem like a strange step if you don't have much of a history with complex numbers.", 'start': 388.173, 'duration': 4.283}, {'end': 397, 'text': 'But describing this distance as a product can be unexpectedly useful.', 'start': 393.177, 'duration': 3.823}, {'end': 403.773, 'text': 'it turns our question into a factoring problem, which is ultimately why patterns among prime numbers are going to come into play.', 'start': 397.768, 'duration': 6.005}, {'end': 408.336, 'text': 'Algebraically, this relation is straightforward enough to verify.', 'start': 404.873, 'duration': 3.463}], 'summary': 'Reflecting over the real axis turns distance into a factoring problem, revealing patterns among prime numbers.', 'duration': 25.627, 'max_score': 382.709, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY382709.jpg'}], 'start': 144.488, 'title': 'Lattice points and circle patterns', 'summary': 'Delves into the concept of lattice points within a circle, discussing the systematic counting of lattice points on given rings and how it is rooted in the distribution of primes, ultimately tying in complex numbers and their relation to the problem.', 'chapters': [{'end': 453.436, 'start': 144.488, 'title': 'Lattice points and circle patterns', 'summary': 'Delves into the concept of lattice points within a circle, discussing the systematic counting of lattice points on given rings and how it is rooted in the distribution of primes, ultimately tying in complex numbers and their relation to the problem.', 'duration': 308.948, 'highlights': ['The systematic counting of lattice points on given rings and its connection to the distribution of primes, is a key concept explored in the chapter.', 'The chapter emphasizes the use of complex numbers and their relation to the problem, including their role in transforming the problem into a factoring problem.', 'The approach of thinking about the 2D plane as the set of all complex numbers is highlighted as a fruitful perspective to consider in addressing questions related to the plane.']}], 'duration': 308.948, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY144488.jpg', 'highlights': ['The systematic counting of lattice points on given rings and its connection to the distribution of primes, is a key concept explored in the chapter.', 'The chapter emphasizes the use of complex numbers and their relation to the problem, including their role in transforming the problem into a factoring problem.', 'The approach of thinking about the 2D plane as the set of all complex numbers is highlighted as a fruitful perspective to consider in addressing questions related to the plane.']}, {'end': 763.978, 'segs': [{'end': 476.811, 'src': 'embed', 'start': 453.436, 'weight': 1, 'content': [{'end': 460.26, 'text': 'and then to stretch out by a factor of 5, which in this case, lands you on the output 25, the square of the magnitude.', 'start': 453.436, 'duration': 6.824}, {'end': 469.766, 'text': 'The collection of all of these lattice points, a plus bi, where a and b are integers, has a special name.', 'start': 463.502, 'duration': 6.264}, {'end': 473.648, 'text': "They're called the Gaussian integers, named after Martin Sheen.", 'start': 470.286, 'duration': 3.362}, {'end': 476.811, 'text': "Geometrically, you'll still be asking the same question.", 'start': 474.409, 'duration': 2.402}], 'summary': 'Stretching by a factor of 5 gives output 25, forming gaussian integers.', 'duration': 23.375, 'max_score': 453.436, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY453436.jpg'}, {'end': 535.874, 'src': 'embed', 'start': 502.462, 'weight': 3, 'content': [{'end': 505.583, 'text': 'for how many lattice points are a given distance away from the origin?', 'start': 502.462, 'duration': 3.121}, {'end': 512.166, 'text': 'To see why, we first need to understand how numbers factor inside the Gaussian integers.', 'start': 506.543, 'duration': 5.623}, {'end': 520.573, 'text': 'As a refresher, among ordinary integers, every number can be factored as some unique collection of prime numbers.', 'start': 513.808, 'duration': 6.765}, {'end': 528.677, 'text': 'For example, 2250 can be factored as 2 times 3 squared, times 5 cubed,', 'start': 521.453, 'duration': 7.224}, {'end': 535.874, 'text': 'and there is no other collection of prime numbers that also multiplies to make 2250..', 'start': 528.677, 'duration': 7.197}], 'summary': 'Exploring factorization of numbers in gaussian integers', 'duration': 33.412, 'max_score': 502.462, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY502462.jpg'}, {'end': 630.368, 'src': 'embed', 'start': 602.575, 'weight': 2, 'content': [{'end': 607.399, 'text': 'This will give you a different way to factor 5 into two distinct Gaussian primes.', 'start': 602.575, 'duration': 4.824}, {'end': 615.347, 'text': 'But other than the things that you can get by multiplying some of these factors by negative 1 or i or negative i,', 'start': 608.46, 'duration': 6.887}, {'end': 618.37, 'text': 'factorization within the Gaussian integers is unique.', 'start': 615.347, 'duration': 3.023}, {'end': 625.885, 'text': 'And if you can figure out how ordinary prime numbers factor inside the Gaussian integers,', 'start': 620.06, 'duration': 5.825}, {'end': 630.368, 'text': "that'll be enough to tell us how any other natural number factors inside these Gaussian integers.", 'start': 625.885, 'duration': 4.483}], 'summary': 'Understanding how prime numbers factor in gaussian integers is crucial for factorization.', 'duration': 27.793, 'max_score': 602.575, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY602575.jpg'}, {'end': 673.718, 'src': 'embed', 'start': 648.718, 'weight': 0, 'content': [{'end': 655.885, 'text': 'This corresponds with the fact that rings with a radius equal to the square root of one of these prime numbers always hit some lattice points.', 'start': 648.718, 'duration': 7.167}, {'end': 660.409, 'text': "In fact, they always hit exactly 8 lattice points, as you'll see in just a moment.", 'start': 656.605, 'duration': 3.804}, {'end': 670.295, 'text': 'On the other hand, prime numbers that are 3 above a multiple of 4, like 3, or 7, or 11,', 'start': 663.809, 'duration': 6.486}, {'end': 673.718, 'text': 'these guys cannot be factored further inside the Gaussian integers.', 'start': 670.295, 'duration': 3.423}], 'summary': 'Rings with radius equal to prime numbers hit 8 lattice points; prime numbers 3 above a multiple of 4 cannot be factored inside gaussian integers.', 'duration': 25, 'max_score': 648.718, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY648718.jpg'}, {'end': 752.415, 'src': 'embed', 'start': 719.656, 'weight': 4, 'content': [{'end': 723.737, 'text': 'The prime number 2, by the way, is a little special because it does factor.', 'start': 719.656, 'duration': 4.081}, {'end': 728.058, 'text': 'You can write it as 1 plus i times 1 minus i.', 'start': 724.017, 'duration': 4.041}, {'end': 731.899, 'text': 'But these two Gaussian primes are a 90 degree rotation away from each other.', 'start': 728.058, 'duration': 3.841}, {'end': 735.6, 'text': 'So you can multiply one of them by i to get the other.', 'start': 732.499, 'duration': 3.101}, {'end': 742.462, 'text': 'And that fact is going to make us want to treat the prime number 2 a little bit differently for where all of this stuff is going.', 'start': 736.5, 'duration': 5.962}, {'end': 744.262, 'text': 'So just keep that in the back of your mind.', 'start': 742.702, 'duration': 1.56}, {'end': 752.415, 'text': 'Remember, our goal here is to count how many lattice points are a given distance away from the origin,', 'start': 746.914, 'duration': 5.501}], 'summary': 'Gaussian primes 1+i & 1-i are 90 degrees apart, influencing treatment of prime number 2.', 'duration': 32.759, 'max_score': 719.656, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY719656.jpg'}], 'start': 453.436, 'title': 'Gaussian integers and lattice points', 'summary': "Explores gaussian integers and lattice points, showing how a factor of 5 stretches lattice points to output 25, and examines the algebraic question of lattice points' distances from the origin. it also emphasizes understanding number factorization inside gaussian integers and discusses the uniqueness and exceptions of factorization within them, highlighting the regularity of prime numbers and their relationship with lattice points.", 'chapters': [{'end': 528.677, 'start': 453.436, 'title': 'Gaussian integers and lattice points', 'summary': 'Explores the concept of gaussian integers, where the collection of lattice points can be stretched by a factor of 5 to land on the output 25, and the question of how many lattice points gaussian integers are a given distance away from the origin, phrased algebraically, is examined, with emphasis on understanding number factorization inside the gaussian integers.', 'duration': 75.241, 'highlights': ['Gaussian integers, where the collection of lattice points can be stretched by a factor of 5 to land on the output 25', 'Question of how many lattice points Gaussian integers are a given distance away from the origin, phrased algebraically', 'Understanding number factorization inside the Gaussian integers']}, {'end': 763.978, 'start': 528.677, 'title': 'Factorization in gaussian integers', 'summary': 'Discusses the uniqueness and exceptions of factorization within the gaussian integers, highlighting the regularity of prime numbers and their relationship with lattice points, with a surprising fact about prime numbers that are 1 above a multiple of 4 and those that are 3 above a multiple of 4.', 'duration': 235.301, 'highlights': ['Prime numbers that are 1 above a multiple of 4 can always be factored into exactly two distinct Gaussian primes, corresponding with the fact that rings with a radius equal to the square root of these prime numbers hit exactly 8 lattice points.', 'Prime numbers that are 3 above a multiple of 4 cannot be factored further inside the Gaussian integers, corresponding with the fact that a ring whose radius is the square root of these primes will never hit any lattice points.', 'The prime number 2 can be factored into 1 plus i times 1 minus i, but these two Gaussian primes are a 90-degree rotation away from each other, making the prime number 2 a little special in the context of factorization within the Gaussian integers.']}], 'duration': 310.542, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY453436.jpg', 'highlights': ['Prime numbers 1 above a multiple of 4 can always be factored into exactly two distinct Gaussian primes, corresponding with the fact that rings with a radius equal to the square root of these prime numbers hit exactly 8 lattice points.', 'Gaussian integers, where the collection of lattice points can be stretched by a factor of 5 to land on the output 25', 'Understanding number factorization inside the Gaussian integers', 'Question of how many lattice points Gaussian integers are a given distance away from the origin, phrased algebraically', 'Prime number 2 can be factored into 1 plus i times 1 minus i, but these two Gaussian primes are a 90-degree rotation away from each other, making the prime number 2 a little special in the context of factorization within the Gaussian integers.', 'Prime numbers 3 above a multiple of 4 cannot be factored further inside the Gaussian integers, corresponding with the fact that a ring whose radius is the square root of these primes will never hit any lattice points.']}, {'end': 1184.698, 'segs': [{'end': 806.425, 'src': 'embed', 'start': 763.978, 'weight': 0, 'content': [{'end': 773.959, 'text': 'is the same as asking how many Gaussian integers have the special property that multiplying them by their complex conjugate gives you 25..', 'start': 763.978, 'duration': 9.981}, {'end': 777.962, 'text': "So here's the recipe for finding all Gaussian integers that have this property.", 'start': 773.959, 'duration': 4.003}, {'end': 784.848, 'text': 'Step 1, factor 25, which inside the ordinary integers looks like 5 squared.', 'start': 778.843, 'duration': 6.005}, {'end': 792.494, 'text': 'but since 5 factors even further, as 2 plus i times 2 minus i, 25 breaks down as these four Gaussian primes.', 'start': 784.848, 'duration': 7.646}, {'end': 798.999, 'text': 'Step 2, organize these into two different columns, with conjugate pairs sitting right next to each other.', 'start': 793.395, 'duration': 5.604}, {'end': 806.425, 'text': "Then once you do that, multiply what's in each column and you'll come out with two different Gaussian integers on the bottom.", 'start': 800.16, 'duration': 6.265}], 'summary': 'Finding gaussian integers with special property, given 25, involves factoring and organizing into columns.', 'duration': 42.447, 'max_score': 763.978, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY763978.jpg'}, {'end': 969.916, 'src': 'embed', 'start': 948.425, 'weight': 2, 'content': [{'end': 958.849, 'text': 'You can either have zero copies of 2 plus i in the left column, one copy in there, two copies in there, or all three of them in that left column.', 'start': 948.425, 'duration': 10.424}, {'end': 969.916, 'text': 'Those four choices, multiplied by the final four choices of multiplying the product from the left column by 1 or by i, or negative, 1 or negative.', 'start': 960.03, 'duration': 9.886}], 'summary': 'Four choices of 2 plus i multiplied by four final choices', 'duration': 21.491, 'max_score': 948.425, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY948425.jpg'}, {'end': 1040.519, 'src': 'embed', 'start': 1014.043, 'weight': 6, 'content': [{'end': 1021.233, 'text': "So for a number like this, 3 times 5 cubed, which is 375, there's actually no lattice point that you'll hit.", 'start': 1014.043, 'duration': 7.19}, {'end': 1028.069, 'text': 'No Gaussian integer whose product with its own conjugate gives you 375.', 'start': 1021.794, 'duration': 6.275}, {'end': 1032.393, 'text': 'However, if you introduce a second factor of 3, then you have an option.', 'start': 1028.069, 'duration': 4.324}, {'end': 1037.096, 'text': 'You can throw one 3 in the left column, and the other 3 in the right column.', 'start': 1032.913, 'duration': 4.183}, {'end': 1040.519, 'text': 'Since 3 is its own complex conjugate.', 'start': 1037.896, 'duration': 2.623}], 'summary': 'For 3 * 5^3 = 375, no lattice point is hit. introducing a second factor of 3 gives an option to place one 3 in each column, as 3 is its complex conjugate.', 'duration': 26.476, 'max_score': 1014.043, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY1014043.jpg'}, {'end': 1117.457, 'src': 'embed', 'start': 1088.978, 'weight': 5, 'content': [{'end': 1094.704, 'text': 'the number of choices they give you will always be one more than the exponent that shows up with that factor.', 'start': 1088.978, 'duration': 5.726}, {'end': 1106.33, 'text': 'On the other hand, for prime factors like 3 or 7 or 11, which are already Gaussian primes and cannot be split if they show up with an even power,', 'start': 1097.185, 'duration': 9.145}, {'end': 1108.772, 'text': 'you have one and only one choice with what to do with them.', 'start': 1106.33, 'duration': 2.442}, {'end': 1113.194, 'text': "But if it's an odd exponent, you're screwed, and you just have zero choices.", 'start': 1109.592, 'duration': 3.602}, {'end': 1117.457, 'text': 'And always, no matter what, you have those final four choices at the end.', 'start': 1114.295, 'duration': 3.162}], 'summary': 'Factors have varying choices based on exponent and primality, concluding with four final choices.', 'duration': 28.479, 'max_score': 1088.978, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY1088978.jpg'}, {'end': 1184.057, 'src': 'heatmap', 'start': 1134.904, 'weight': 4, 'content': [{'end': 1140.026, 'text': 'The one last thing to mention about this recipe is how factors of two affect the count.', 'start': 1134.904, 'duration': 5.122}, {'end': 1148.211, 'text': 'If your number is even, then that factor of 2 breaks down as 1 plus i times 1 minus i.', 'start': 1141.144, 'duration': 7.067}, {'end': 1151.774, 'text': 'So you can divvy up that complex conjugate pair between the two columns.', 'start': 1148.211, 'duration': 3.563}, {'end': 1160.482, 'text': 'And at first, it might look like this doubles your options, depending on how you choose to place those two Gaussian primes between the columns.', 'start': 1152.715, 'duration': 7.767}, {'end': 1168.928, 'text': 'However, since multiplying one of these guys by i gives you the other one when you swap them between the columns,', 'start': 1161.363, 'duration': 7.565}, {'end': 1176.192, 'text': 'the effect that that has on the output from the left column is to just multiply it by i or by negative i.', 'start': 1168.928, 'duration': 7.264}, {'end': 1184.057, 'text': "So that's actually redundant, with the final step where we take the product of this left column and choose to multiply it either by 1, i, negative 1,", 'start': 1176.192, 'duration': 7.865}], 'summary': 'Factors of two affect count in complex conjugates, doubling options but leading to redundancy.', 'duration': 25.578, 'max_score': 1134.904, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY1134904.jpg'}], 'start': 763.978, 'title': 'Gaussian integer factorization and counting lattice points', 'summary': 'Explains a method to find all gaussian integers whose product with their complex conjugate is 25, demonstrating the process and its application with different examples like 125. additionally, it discusses the method of counting lattice points on circles with a radius square root of n, highlighting the impact of different factors on the number of lattice points and the choices available for divvying up the factors.', 'chapters': [{'end': 969.916, 'start': 763.978, 'title': 'Gaussian integer factorization', 'summary': 'Explains a method to find all gaussian integers whose product with their complex conjugate is 25, demonstrating the process and its application with different examples like 125.', 'duration': 205.938, 'highlights': ['The recipe for finding all Gaussian integers that have the property of multiplying them by their complex conjugate gives you 25.', 'Factorizing 25 into Gaussian primes and organizing them into two different columns with conjugate pairs to find different Gaussian integers.', 'Explaining the different choices for dividing the primes into columns and the final step of choosing to multiply the product from the left column by 1, i, negative 1, or negative i.']}, {'end': 1184.698, 'start': 969.916, 'title': 'Counting lattice points', 'summary': 'Discusses the method of counting lattice points on circles with a radius square root of n, highlighting the impact of different factors on the number of lattice points and the choices available for divvying up the factors.', 'duration': 214.782, 'highlights': ['When factoring n, prime numbers like 5, 13, and 17, which further factor into a complex conjugate pair of Gaussian primes, provide one more choice than the exponent that shows up with that factor.', "For prime factors like 3, 7, and 11, if they show up with an even power, there is only one choice, but if it's an odd exponent, there are zero choices.", 'Factors of two affect the count by breaking down as 1 plus i times 1 minus i, allowing the divvying up of the complex conjugate pair between the two columns, potentially doubling the options.', "Introducing a factor like 3 that doesn't break down as the product of two conjugate Gaussian primes leads to imbalance and no lattice points hit, but adding a second factor of 3 provides options for balanced columns and hitting lattice points."]}], 'duration': 420.72, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY763978.jpg', 'highlights': ['Factorizing 25 into Gaussian primes and organizing them into two different columns with conjugate pairs to find different Gaussian integers.', 'The recipe for finding all Gaussian integers that have the property of multiplying them by their complex conjugate gives you 25.', 'Explaining the different choices for dividing the primes into columns and the final step of choosing to multiply the product from the left column by 1, i, negative 1, or negative i.', 'When factoring n, prime numbers like 5, 13, and 17, which further factor into a complex conjugate pair of Gaussian primes, provide one more choice than the exponent that shows up with that factor.', 'Factors of two affect the count by breaking down as 1 plus i times 1 minus i, allowing the divvying up of the complex conjugate pair between the two columns, potentially doubling the options.', "For prime factors like 3, 7, and 11, if they show up with an even power, there is only one choice, but if it's an odd exponent, there are zero choices.", "Introducing a factor like 3 that doesn't break down as the product of two conjugate Gaussian primes leads to imbalance and no lattice points hit, but adding a second factor of 3 provides options for balanced columns and hitting lattice points."]}, {'end': 1762.508, 'segs': [{'end': 1234.996, 'src': 'embed', 'start': 1186.579, 'weight': 0, 'content': [{'end': 1193.181, 'text': "What this means is that a factor of 2, or any power of 2, doesn't actually change the count at all.", 'start': 1186.579, 'duration': 6.602}, {'end': 1195.502, 'text': "It doesn't hurt, but it doesn't help.", 'start': 1193.721, 'duration': 1.781}, {'end': 1200.744, 'text': 'For example, a circle with radius √5 hits 8 lattice points.', 'start': 1196.362, 'duration': 4.382}, {'end': 1204.025, 'text': 'And if you grow that radius to √10, then you also hit 8 lattice points.', 'start': 1201.244, 'duration': 2.781}, {'end': 1206.326, 'text': 'And √20 also hits 8 lattice points, as does √40.', 'start': 1204.125, 'duration': 2.201}, {'end': 1207.686, 'text': "Factors of 2 just don't make a difference.", 'start': 1206.326, 'duration': 1.36}, {'end': 1218.293, 'text': "Now what's about to happen is number theory at its best.", 'start': 1215.612, 'duration': 2.681}, {'end': 1225.034, 'text': 'We have this complicated recipe telling us how many lattice points sit on a circle with radius square root of n,', 'start': 1218.933, 'duration': 6.101}, {'end': 1228.555, 'text': 'and it depends on the prime factorization of n.', 'start': 1225.034, 'duration': 3.521}, {'end': 1234.996, 'text': "To turn this into something simpler, something we can actually deal with, we're going to exploit the regularity of primes,", 'start': 1228.555, 'duration': 6.441}], 'summary': "A factor of 2 or any power of 2 doesn't change the lattice point count on a circle, which depends on the prime factorization of its radius.", 'duration': 48.417, 'max_score': 1186.579, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY1186579.jpg'}, {'end': 1346.141, 'src': 'embed', 'start': 1313.237, 'weight': 2, 'content': [{'end': 1316.719, 'text': 'So for our central question of counting lattice points in this way.', 'start': 1313.237, 'duration': 3.482}, {'end': 1318.84, 'text': 'that involves factoring a number.', 'start': 1316.719, 'duration': 2.121}, {'end': 1326.604, 'text': "what I'm going to do is write down the number of choices we have, but using chi in what at first seems like a much more complicated way,", 'start': 1318.84, 'duration': 7.764}, {'end': 1329.485, 'text': 'but this has the benefit of treating all prime factors equally.', 'start': 1326.604, 'duration': 2.881}, {'end': 1339.014, 'text': 'For each prime power, like 5 cubed, what you write down is chi of 1 plus chi of 5 plus chi of 5 squared plus chi of 5 cubed.', 'start': 1330.606, 'duration': 8.408}, {'end': 1346.141, 'text': 'You add up the value of chi on all the powers of this prime up to the one that shows up inside the factorization.', 'start': 1339.635, 'duration': 6.506}], 'summary': 'Count lattice points by factoring number and using chi in a more complicated way to treat all prime factors equally.', 'duration': 32.904, 'max_score': 1313.237, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY1313237.jpg'}, {'end': 1674.328, 'src': 'embed', 'start': 1648.949, 'weight': 3, 'content': [{'end': 1657.996, 'text': 'what it means is that the total number of lattice points inside this big circle is approximately 4 times r squared times this sum.', 'start': 1648.949, 'duration': 9.047}, {'end': 1668.685, 'text': 'And because chi is 0 on every even number and it oscillates between 1 and negative 1 for odd numbers, this sum looks like 1, minus 1,', 'start': 1658.637, 'duration': 10.048}, {'end': 1672.088, 'text': 'third plus a fifth, minus 1, seventh and so on.', 'start': 1668.685, 'duration': 3.403}, {'end': 1674.328, 'text': 'And this is exactly what we wanted.', 'start': 1672.967, 'duration': 1.361}], 'summary': 'The total number of lattice points inside the circle is approximately 4 times r squared, and the sum alternates between 1 and -1 for odd numbers.', 'duration': 25.379, 'max_score': 1648.949, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY1648949.jpg'}], 'start': 1186.579, 'title': 'Counting lattice points', 'summary': "Discusses the count of lattice points on a circle with radius √n, showing that a factor of 2 doesn't change the count, and the count depends on the prime factorization of n. it also explores the properties of the multiplicative function chi, its role in factoring numbers, and its application in counting the lattice points inside a big circle, leading to an alternate expression for the total number of lattice points and its connection to pi.", 'chapters': [{'end': 1234.996, 'start': 1186.579, 'title': 'Circle lattice points', 'summary': "Discusses how the count of lattice points on a circle with radius √n is affected by factors of 2 and the prime factorization of n, showcasing that a factor of 2 doesn't change the count, and the count depends on the prime factorization of n.", 'duration': 48.417, 'highlights': ['The count of lattice points on a circle with radius √n is unaffected by factors of 2, as demonstrated by examples like √5, √10, √20, and √40 all hitting 8 lattice points.', 'The number of lattice points on a circle with radius square root of n is determined by the prime factorization of n, showcasing the intricate nature of number theory.']}, {'end': 1762.508, 'start': 1234.996, 'title': 'Gaussian integers and multiplicative functions', 'summary': 'Explores the properties of the multiplicative function chi, its role in factoring numbers, and its application in counting the lattice points inside a big circle, leading to an alternate expression for the total number of lattice points and its connection to pi.', 'duration': 527.512, 'highlights': ['The role of the multiplicative function chi in factoring numbers and counting lattice points inside a big circle is explored, leading to an alternate expression for the total number of lattice points and its connection to pi.', 'The properties of the multiplicative function chi and its application in counting lattice points inside a big circle are discussed, illustrating a connection to pi and its convergence to pi as r increases.', 'Examination of the organized expression for the total number of lattice points inside a big circle and its approximation using the multiplicative function chi, leading to the convergence of the infinite sum to pi.']}], 'duration': 575.929, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/NaL_Cb42WyY/pics/NaL_Cb42WyY1186579.jpg', 'highlights': ['The count of lattice points on a circle with radius √n is unaffected by factors of 2, as demonstrated by examples like √5, √10, √20, and √40 all hitting 8 lattice points.', 'The number of lattice points on a circle with radius square root of n is determined by the prime factorization of n, showcasing the intricate nature of number theory.', 'The role of the multiplicative function chi in factoring numbers and counting lattice points inside a big circle is explored, leading to an alternate expression for the total number of lattice points and its connection to pi.', 'Examination of the organized expression for the total number of lattice points inside a big circle and its approximation using the multiplicative function chi, leading to the convergence of the infinite sum to pi.', 'The properties of the multiplicative function chi and its application in counting lattice points inside a big circle are discussed, illustrating a connection to pi and its convergence to pi as r increases.']}], 'highlights': ['The formula for pi is derived from a certain regularity in the behavior of prime numbers when placed inside the complex numbers, providing a deeper understanding of the circle hidden behind it.', 'The count of lattice points on a circle with radius √n is unaffected by factors of 2, as demonstrated by examples like √5, √10, √20, and √40 all hitting 8 lattice points.', 'Prime numbers 1 above a multiple of 4 can always be factored into exactly two distinct Gaussian primes, corresponding with the fact that rings with a radius equal to the square root of these prime numbers hit exactly 8 lattice points.', 'The role of the multiplicative function chi in factoring numbers and counting lattice points inside a big circle is explored, leading to an alternate expression for the total number of lattice points and its connection to pi.', 'The systematic counting of lattice points on given rings and its connection to the distribution of primes, is a key concept explored in the chapter.']}