title

Why do prime numbers make these spirals? | Dirichlet’s theorem and pi approximations

description

A curious pattern, approximations for pi, and prime distributions.
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An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: http://3b1b.co/spiral-thanks
Based on this Math Stack Exchange post:
https://math.stackexchange.com/questions/885879/meaning-of-rays-in-polar-plot-of-prime-numbers/885894
Want to learn more about rational approximations? See this Mathologer video.
https://youtu.be/CaasbfdJdJg
Also, if you haven't heard of Ulam Spirals, you may enjoy this Numberphile video:
https://youtu.be/iFuR97YcSLM
Dirichlet's paper:
https://arxiv.org/pdf/0808.1408.pdf
Timestamps:
0:00 - The spiral mystery
3:35 - Non-prime spirals
6:10 - Residue classes
7:20 - Why the galactic spirals
9:30 - Euler’s totient function
10:28 - The larger scale
14:45 - Dirichlet’s theorem
20:26 - Why care?
Corrections:
18:30: In the video, I say that Dirichlet showed that the primes are equally distributed among allowable residue classes, but this is not historically accurate. (By "allowable", here, I mean a residue class whose elements are coprime to the modulus, as described in the video). What he actually showed is that the sum of the reciprocals of all primes in a given allowable residue class diverges, which proves that there are infinitely many primes in such a sequence.
Dirichlet observed this equal distribution numerically and noted this in his paper, but it wasn't until decades later that this fact was properly proved, as it required building on some of the work of Riemann in his famous 1859 paper. If I'm not mistaken, I think it wasn't until Vallée Poussin in (1899), with a version of the prime number theorem for residue classes like this, but I could be wrong there.
In many ways, this was a very silly error for me to have let through. It is true that this result was proven with heavy use of complex analysis, and in fact, it's in a complex analysis lecture that I remember first learning about it. But of course, this would have to have happened after Dirichlet because it would have to have happened after Riemann!
My apologies for the mistake. If you notice factual errors in videos that are not already mentioned in the video's description or pinned comment, don't hesitate to let me know.
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If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
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detail

{'title': 'Why do prime numbers make these spirals? | Dirichlet’s theorem and pi approximations', 'heatmap': [{'end': 1341, 'start': 1327.59, 'weight': 1}], 'summary': "Explores the relationship between prime numbers and polar coordinates, revealing patterns in archimedean spirals and residue classes mod 6 and 10. it discusses the significance of dirichlet's theorem, rational approximations for pi, and the distribution of primes among residue classes, emphasizing the value of independent exploration in mathematics.", 'chapters': [{'end': 109.101, 'segs': [{'end': 43.821, 'src': 'embed', 'start': 4.218, 'weight': 0, 'content': [{'end': 8.402, 'text': "I first saw this pattern that I'm about to show you in a question on the Math Stack Exchange.", 'start': 4.218, 'duration': 4.184}, {'end': 13.166, 'text': 'It was asked by a user under the name Dwimark and answered by Greg Martin,', 'start': 8.902, 'duration': 4.264}, {'end': 17.69, 'text': 'and it relates to the distribution of prime numbers together with rational approximations for pi.', 'start': 13.166, 'duration': 4.524}, {'end': 22.034, 'text': 'You see, what the user had been doing was playing around with data in polar coordinates.', 'start': 18.43, 'duration': 3.604}, {'end': 24.955, 'text': "As a quick reminder, so we're all on the same page.", 'start': 22.674, 'duration': 2.281}, {'end': 34.678, 'text': 'this means labeling points in 2D space, not with the usual x-y coordinates, but instead with a distance from the origin, commonly called r for radius,', 'start': 24.955, 'duration': 9.723}, {'end': 39.52, 'text': 'together with the angle that that radial line makes with the horizontal, commonly called theta.', 'start': 34.678, 'duration': 4.842}, {'end': 43.821, 'text': 'And for our purposes, this angle will be measured in radians,', 'start': 40.32, 'duration': 3.501}], 'summary': 'Pattern in prime numbers and rational approximations for pi in polar coordinates', 'duration': 39.603, 'max_score': 4.218, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ4218.jpg'}, {'end': 96.973, 'src': 'embed', 'start': 67.711, 'weight': 2, 'content': [{'end': 69.773, 'text': 'There is no practical reason to do this.', 'start': 67.711, 'duration': 2.062}, {'end': 70.913, 'text': "It's purely fun.", 'start': 70.093, 'duration': 0.82}, {'end': 74.056, 'text': "We're just frolicking around in the playground of data visualization.", 'start': 71.094, 'duration': 2.962}, {'end': 78.399, 'text': 'And to get a sense for what it means, look at all the whole numbers rather than just the primes.', 'start': 74.596, 'duration': 3.803}, {'end': 87.826, 'text': 'The point sits a distance one away from the origin with an angle of one radian, which actually means this arc is the same length as that radial line.', 'start': 78.999, 'duration': 8.827}, {'end': 92.489, 'text': 'And then has twice that angle and twice the distance.', 'start': 88.606, 'duration': 3.883}, {'end': 96.973, 'text': 'And to get to, you rotate one more radian with a total angle.', 'start': 93.21, 'duration': 3.763}], 'summary': 'Exploring data visualization for fun, using whole numbers and angles, with a total rotation of one more radian.', 'duration': 29.262, 'max_score': 67.711, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ67711.jpg'}], 'start': 4.218, 'title': 'Prime numbers in polar coordinates', 'summary': 'Explores the pattern of plotting points in polar coordinates with prime numbers, discovered on math stack exchange. it relates to prime number distribution, rational approximations for pi, and unique properties of polar coordinates.', 'chapters': [{'end': 109.101, 'start': 4.218, 'title': 'Prime numbers in polar coordinates', 'summary': 'Explores the pattern of plotting points in polar coordinates where both coordinates are prime numbers, which was discovered on the math stack exchange. it relates to the distribution of prime numbers and rational approximations for pi, with a focus on the unique properties of polar coordinates.', 'duration': 104.883, 'highlights': ['The pattern involves plotting points in polar coordinates where both coordinates are prime numbers.', 'Polar coordinates are used to label points in 2D space with a distance from the origin and an angle measured in radians.', 'The chapter emphasizes that there is no practical reason for this exploration, as it is purely for fun and data visualization.']}], 'duration': 104.883, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ4218.jpg', 'highlights': ['The pattern involves plotting points in polar coordinates where both coordinates are prime numbers.', 'Polar coordinates are used to label points in 2D space with a distance from the origin and an angle measured in radians.', 'The chapter emphasizes that there is no practical reason for this exploration, as it is purely for fun and data visualization.']}, {'end': 357.168, 'segs': [{'end': 154.737, 'src': 'embed', 'start': 126.193, 'weight': 2, 'content': [{'end': 132.579, 'text': 'Now if you make the admittedly arbitrary move to knock out everything except the prime numbers, it initially looks quite random.', 'start': 126.193, 'duration': 6.386}, {'end': 136.522, 'text': 'After all, primes are famous for their chaotic and difficult to predict behavior.', 'start': 133.019, 'duration': 3.503}, {'end': 145.329, 'text': 'But when you zoom out, what you start to see are these very clear, galactic-seeming spirals.', 'start': 137.142, 'duration': 8.187}, {'end': 148.392, 'text': "And what's weird is some of the arms seem to be missing.", 'start': 146.15, 'duration': 2.242}, {'end': 154.737, 'text': 'And zooming out even further, those spirals give way to a different pattern.', 'start': 150.653, 'duration': 4.084}], 'summary': 'Zooming out from prime numbers reveals clear spirals and a different pattern.', 'duration': 28.544, 'max_score': 126.193, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ126193.jpg'}, {'end': 275.596, 'src': 'embed', 'start': 248.53, 'weight': 0, 'content': [{'end': 252.392, 'text': "But don't be too disappointed, because both these questions are still phenomenal puzzles.", 'start': 248.53, 'duration': 3.862}, {'end': 257.735, 'text': "There's a very satisfying answer for the spirals, and even if the primes don't cause the spirals,", 'start': 252.773, 'duration': 4.962}, {'end': 265.06, 'text': 'asking what goes on when you filter for those primes does lead you to one of the most important theorems about the distribution of prime numbers,', 'start': 257.735, 'duration': 7.325}, {'end': 267.061, 'text': "known in number theory as Dirichlet's theorem.", 'start': 265.06, 'duration': 2.001}, {'end': 271.373, 'text': "To kick things off, let's zoom back in a little bit smaller.", 'start': 268.991, 'duration': 2.382}, {'end': 275.596, 'text': 'Did you notice that, as we were zooming out, there were these six little spirals?', 'start': 272.093, 'duration': 3.503}], 'summary': "The questions about spirals and primes lead to dirichlet's theorem.", 'duration': 27.066, 'max_score': 248.53, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ248530.jpg'}, {'end': 345.736, 'src': 'embed', 'start': 316.586, 'weight': 1, 'content': [{'end': 318.746, 'text': 'Another six steps, a slightly smaller angle.', 'start': 316.586, 'duration': 2.16}, {'end': 322.867, 'text': 'Six more steps, smaller still, and so on.', 'start': 320.126, 'duration': 2.741}, {'end': 328.008, 'text': 'With this angle changing gently enough that it gives the illusion of a single curving line.', 'start': 323.367, 'duration': 4.641}, {'end': 334.227, 'text': "When you limit the view to prime numbers, all but two of these spiral arms, they'll go away.", 'start': 329.403, 'duration': 4.824}, {'end': 335.748, 'text': 'And think about it.', 'start': 334.948, 'duration': 0.8}, {'end': 338.951, 'text': "A prime number can't be a multiple of 6.", 'start': 336.189, 'duration': 2.762}, {'end': 345.736, 'text': "And it also can't be 2 above a multiple of 6, unless it's 2, or 4 above a multiple of 6, since all of those are even numbers.", 'start': 338.951, 'duration': 6.785}], 'summary': 'Prime numbers form a spiral with 2 arms, disappearing for multiples of 6.', 'duration': 29.15, 'max_score': 316.586, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ316586.jpg'}], 'start': 109.682, 'title': 'Prime numbers and archimedean spirals', 'summary': "Explores the emergence of archimedean spirals and patterns when zooming out from prime numbers, revealing 20 spirals and 280 rays, leading to the important theorem of dirichlet's theorem in number theory.", 'chapters': [{'end': 357.168, 'start': 109.682, 'title': 'Prime numbers and archimedean spirals', 'summary': "Explores the emergence of archimedean spirals and patterns when zooming out from prime numbers, revealing 20 spirals and 280 rays, leading to the important theorem of dirichlet's theorem in number theory.", 'duration': 247.486, 'highlights': ["The emergence of Archimedean spirals and patterns when zooming out from prime numbers, revealing 20 spirals and 280 rays, leads to the important theorem of Dirichlet's theorem in number theory.", "The explanation of the spirals' formation using multiples of six, where each step forward involves a turn of one radian, resulting in the illusion of a single curving line.", 'The insight that when limiting the view to prime numbers, all but two of the spiral arms disappear, indicating that nothing magical is occurring at this smaller scale.']}], 'duration': 247.486, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ109682.jpg', 'highlights': ["The emergence of Archimedean spirals and patterns when zooming out from prime numbers, revealing 20 spirals and 280 rays, leads to the important theorem of Dirichlet's theorem in number theory.", "The explanation of the spirals' formation using multiples of six, where each step forward involves a turn of one radian, resulting in the illusion of a single curving line.", 'The insight that when limiting the view to prime numbers, all but two of the spiral arms disappear, indicating that nothing magical is occurring at this smaller scale.']}, {'end': 627.97, 'segs': [{'end': 420.793, 'src': 'embed', 'start': 379.095, 'weight': 3, 'content': [{'end': 385.515, 'text': 'So, for example, 6 goes into 20 three times and it leaves a remainder of 2.', 'start': 379.095, 'duration': 6.42}, {'end': 390.019, 'text': 'So 20 has a residue of 2 mod 6.', 'start': 385.515, 'duration': 4.504}, {'end': 398.386, 'text': 'Together with all the other numbers leaving a remainder of 2 when the thing you divide by is 6, you have a full residue class mod 6.', 'start': 390.019, 'duration': 8.367}, {'end': 404.111, 'text': "I know that that sounds like the world's most pretentious way of saying everything 2 above a multiple of 6,", 'start': 398.386, 'duration': 5.725}, {'end': 407.994, 'text': 'but this is the standard jargon and it is actually handy to have some words for the idea.', 'start': 404.111, 'duration': 3.883}, {'end': 415.689, 'text': 'So, looking at our diagram in the lingo, each of these spiral arms corresponds to a residue class, mod6,', 'start': 408.823, 'duration': 6.866}, {'end': 418.812, 'text': 'and the reason we see them is that 6 is close to 2 pi.', 'start': 415.689, 'duration': 3.123}, {'end': 420.793, 'text': 'turning 6 radians is almost a full turn.', 'start': 418.812, 'duration': 1.981}], 'summary': 'Explaining residue classes mod 6 using 20 and 2 as an example.', 'duration': 41.698, 'max_score': 379.095, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ379095.jpg'}, {'end': 517.416, 'src': 'embed', 'start': 492.677, 'weight': 0, 'content': [{'end': 498.238, 'text': 'Same for everything 2 above a multiple of 44, and so on, eventually filling out the full diagram.', 'start': 492.677, 'duration': 5.561}, {'end': 501.419, 'text': 'To phrase it with our fancier language.', 'start': 499.319, 'duration': 2.1}, {'end': 512.75, 'text': 'each of these spiral arms shows a residue class mod 44, And maybe now you can tell me what happens when we limit our view to prime numbers.', 'start': 501.419, 'duration': 11.331}, {'end': 517.416, 'text': "Primes cannot be a multiple of 44, so that arm won't be visible.", 'start': 513.751, 'duration': 3.665}], 'summary': "Residue classes mod 44 fill diagram; prime numbers don't show as a multiple of 44.", 'duration': 24.739, 'max_score': 492.677, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ492677.jpg'}, {'end': 571.354, 'src': 'embed', 'start': 541.346, 'weight': 1, 'content': [{'end': 543.889, 'text': 'This is what gives the picture those Milky Way-seeming gaps.', 'start': 541.346, 'duration': 2.543}, {'end': 550.663, 'text': "Each spiral we're left with is a residue class that doesn't share any prime factors with 44.", 'start': 544.749, 'duration': 5.914}, {'end': 556.506, 'text': "And within each one of those arms that we can't reject out of hand, the prime numbers seem to be sort of randomly distributed.", 'start': 550.663, 'duration': 5.843}, {'end': 558.827, 'text': "And that's a fact that I'd like you to tuck away.", 'start': 557.006, 'duration': 1.821}, {'end': 559.948, 'text': "We'll return to it later.", 'start': 558.987, 'duration': 0.961}, {'end': 565.011, 'text': 'This is another good chance to inject some of the jargon that mathematicians use.', 'start': 561.589, 'duration': 3.422}, {'end': 571.354, 'text': "What we care about right here are all the numbers between 0 and 43 that don't share a prime factor with 44, right?", 'start': 565.531, 'duration': 5.823}], 'summary': "Prime numbers in residue classes don't share factors with 44, forming randomly distributed patterns.", 'duration': 30.008, 'max_score': 541.346, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ541346.jpg'}, {'end': 608.798, 'src': 'embed', 'start': 583.509, 'weight': 2, 'content': [{'end': 590.413, 'text': 'In this example, you could count that there are 20 different numbers between 1 and 44 that are co-prime to 44.', 'start': 583.509, 'duration': 6.904}, {'end': 596.593, 'text': 'And this is a fact that a number theorist would compactly write by saying phi of 44 equals 20.', 'start': 590.413, 'duration': 6.18}, {'end': 602.556, 'text': "where the Greek letter phi here refers to Euler's totient function, yet another needlessly fancy word,", 'start': 596.593, 'duration': 5.963}, {'end': 608.798, 'text': 'which is defined to be the number of integers from 1 up to n, which are co-primed to n.', 'start': 602.556, 'duration': 6.242}], 'summary': 'There are 20 numbers co-prime to 44, written as phi of 44 equals 20.', 'duration': 25.289, 'max_score': 583.509, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ583509.jpg'}], 'start': 357.788, 'title': 'Residue classes mod 6 and prime spiral visualization', 'summary': "Discusses residue classes mod6, particularly focusing on numbers leaving remainder 2 when divided by 6 forming a full residue class, and explains prime spiral visualization, emphasizing the concept of co-prime numbers and euler's totient function.", 'chapters': [{'end': 420.793, 'start': 357.788, 'title': 'Residue classes mod 6', 'summary': 'Discusses residue classes mod6, where numbers leaving a remainder of 2 when divided by 6 form a full residue class, and explains how each spiral arm corresponds to a residue class, mod6, due to 6 being close to 2 pi.', 'duration': 63.005, 'highlights': ['Each of these spiral arms corresponds to a residue class, mod6, and the reason we see them is that 6 is close to 2 pi, turning 6 radians is almost a full turn.', '6 goes into 20 three times and it leaves a remainder of 2, so 20 has a residue of 2 mod 6.']}, {'end': 627.97, 'start': 421.514, 'title': 'Prime spiral visualization', 'summary': "Explains the visualization of prime numbers through a spiral diagram, showing the residue classes mod 44 and the concept of co-prime numbers, with a specific focus on the euler's totient function and its application, demonstrating the visualization of prime numbers and co-prime numbers.", 'duration': 206.456, 'highlights': ["The Euler's totient function, phi of 44, equals 20, representing the number of integers between 1 and 44 that are co-prime to 44, demonstrating the concept of co-prime numbers.", 'The visualization of prime numbers through a spiral diagram shows the residue classes mod 44, revealing the gaps and pattern of prime numbers within each residue class.', 'The explanation of how prime numbers form gaps in the visualization due to their relation with the residue classes of 44, highlighting the concept of residue classes and its impact on the distribution of prime numbers.', 'The concept of residue classes mod 44 illustrates the distribution of prime numbers and co-prime numbers within the spiral diagram, demonstrating the relationship between prime numbers and residue classes.']}], 'duration': 270.182, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ357788.jpg', 'highlights': ['The concept of residue classes mod 44 illustrates the distribution of prime numbers and co-prime numbers within the spiral diagram, demonstrating the relationship between prime numbers and residue classes.', 'The explanation of how prime numbers form gaps in the visualization due to their relation with the residue classes of 44, highlighting the concept of residue classes and its impact on the distribution of prime numbers.', "The Euler's totient function, phi of 44, equals 20, representing the number of integers between 1 and 44 that are co-prime to 44, demonstrating the concept of co-prime numbers.", '6 goes into 20 three times and it leaves a remainder of 2, so 20 has a residue of 2 mod 6.', 'Each of these spiral arms corresponds to a residue class, mod6, and the reason we see them is that 6 is close to 2 pi, turning 6 radians is almost a full turn.']}, {'end': 982.825, 'segs': [{'end': 693.092, 'src': 'embed', 'start': 628.531, 'weight': 0, 'content': [{'end': 636.822, 'text': 'The first is that 44 sevenths is a very close rational approximation for 2 pi, which results in the residue classes, mod 44,', 'start': 628.531, 'duration': 8.291}, {'end': 638.324, 'text': 'being cleanly separated out.', 'start': 636.822, 'duration': 1.502}, {'end': 645.693, 'text': "The second is that many of these residue classes contain zero prime numbers, or sometimes just one, so they won't show up.", 'start': 639.83, 'duration': 5.863}, {'end': 653.176, 'text': 'But on the other hand, primes do show up plentifully enough in all 20 of the other residue classes that it makes these spiral arms visible.', 'start': 646.193, 'duration': 6.983}, {'end': 657.97, 'text': "And at this point, maybe you can predict what's going on at the larger scale.", 'start': 654.727, 'duration': 3.243}, {'end': 666.297, 'text': 'Just as 6 radians is vaguely close to a full turn and 44 radians is quite close to 7 full turns,', 'start': 659.551, 'duration': 6.746}, {'end': 672.003, 'text': 'it just so happens that 710 radians is extremely close to a whole number of full turns.', 'start': 666.297, 'duration': 5.706}, {'end': 678.789, 'text': "Visually, you can see this by the fact that the point ends up almost exactly on the x-axis, but it's more compelling analytically.", 'start': 672.683, 'duration': 6.106}, {'end': 691.431, 'text': '710 radians is 710 divided by 2 pi rotations, which works out to be 113.000095.', 'start': 679.569, 'duration': 11.862}, {'end': 693.092, 'text': 'Some of you may have seen this in another form.', 'start': 691.431, 'duration': 1.661}], 'summary': '44 sevenths closely approximates 2 pi, making spiral arms visible in 20 residue classes.', 'duration': 64.561, 'max_score': 628.531, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ628531.jpg'}, {'end': 767.763, 'src': 'embed', 'start': 735.808, 'weight': 2, 'content': [{'end': 739.251, 'text': 'Even very far out, one of these sequences looks like a straight line.', 'start': 735.808, 'duration': 3.443}, {'end': 746.718, 'text': 'And of course, the other residue classes mod710 also form these nearly straight lines.', 'start': 740.092, 'duration': 6.626}, {'end': 752.883, 'text': "710 is a big number though, so when all of them are on screen, and there's only so many pixels on the screen, it's a little hard to make them out.", 'start': 746.878, 'duration': 6.005}, {'end': 761.021, 'text': "So in this case, it's actually easier to see when we limit the view to primes, where you don't see many of those residue classes.", 'start': 754.778, 'duration': 6.243}, {'end': 767.763, 'text': 'In reality, with a little further zooming, you can see that there actually is a very gentle spiral to these.', 'start': 762.141, 'duration': 5.622}], 'summary': 'Residue classes mod 710 form nearly straight lines, easier to see when limited to primes.', 'duration': 31.955, 'max_score': 735.808, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ735808.jpg'}], 'start': 628.531, 'title': 'Rational approximations and residue classes', 'summary': 'Discusses rational approximation of 2 pi as 44 sevenths, residue classes mod 44, visibility of spiral arms due to primes, and the significance of 710 radians being extremely close to a whole number of rotations. it also explores residue classes mod710 forming nearly straight lines, the significance of 280 in relation to prime numbers, and the distribution of primes among residue classes mod10.', 'chapters': [{'end': 733.119, 'start': 628.531, 'title': 'Rational approximations and spirals', 'summary': 'Discusses the rational approximation of 2 pi as 44 sevenths, the separation of residue classes mod 44, the visibility of spiral arms due to the presence of primes, and the significance of 710 radians being extremely close to a whole number of full turns, represented as 113.000095 rotations, implying that the angles of new points are almost exactly the same as the last one when moving forward by steps of 710.', 'duration': 104.588, 'highlights': ['The significance of 710 radians being extremely close to a whole number of full turns, visually represented by the point almost exactly on the x-axis and analytically as 710 divided by 2 pi rotations being 113.000095, implying that the angle of each new point is almost exactly the same as the last one when moving forward by steps of 710.', 'The rational approximation of 2 pi as 44 sevenths, leading to the clean separation of residue classes mod 44.', 'The visibility of spiral arms due to the presence of primes in all 20 of the other residue classes, making them plentifully show up.', 'The discussion on rational approximations for 2 pi, particularly the mention of 355 over 113 as a very good approximation for pi and its unusual significance compared to famous irrationals like phi, e, or square root of 2.']}, {'end': 982.825, 'start': 735.808, 'title': 'Residue classes mod710', 'summary': 'Discusses residue classes mod710, illustrating how they form nearly straight lines, explaining the significance of 280 in relation to prime numbers, and delving into the distribution of primes among residue classes mod10.', 'duration': 247.017, 'highlights': ['Residue classes mod710 form nearly straight lines, illustrating a gentle spiral with further zooming, serving as a great approximation for 2 pi.', "The significance of 280 in relation to prime numbers, representing the count of numbers from 1 to 710 that don't share any prime factors with 710.", 'The distribution of primes among residue classes mod10, showing a fairly even spread of about 25% each for the four classes.']}], 'duration': 354.294, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ628531.jpg', 'highlights': ['The significance of 710 radians being extremely close to a whole number of full turns, visually represented by the point almost exactly on the x-axis and analytically as 710 divided by 2 pi rotations being 113.000095, implying that the angle of each new point is almost exactly the same as the last one when moving forward by steps of 710.', 'The rational approximation of 2 pi as 44 sevenths, leading to the clean separation of residue classes mod 44.', 'Residue classes mod710 form nearly straight lines, illustrating a gentle spiral with further zooming, serving as a great approximation for 2 pi.', 'The visibility of spiral arms due to the presence of primes in all 20 of the other residue classes, making them plentifully show up.']}, {'end': 1316.999, 'segs': [{'end': 1010.096, 'src': 'embed', 'start': 983.566, 'weight': 0, 'content': [{'end': 993.155, 'text': 'If you look at all the prime numbers less than some big number x, and you consider what fraction of them are, say 1 above a multiple of 10,', 'start': 983.566, 'duration': 9.589}, {'end': 994.876, 'text': 'that fraction should approach 1.', 'start': 993.155, 'duration': 1.721}, {'end': 997.099, 'text': 'fourth, as x approaches infinity.', 'start': 994.876, 'duration': 2.223}, {'end': 1005.854, 'text': 'And likewise for all of the other allowable residue classes, like 3 and 7 and 9.', 'start': 999.241, 'duration': 6.613}, {'end': 1007.915, 'text': "Of course, there's nothing special about 10.", 'start': 1005.854, 'duration': 2.061}, {'end': 1010.096, 'text': 'A similar fact should hold for any other number.', 'start': 1007.915, 'duration': 2.181}], 'summary': 'As x approaches infinity, the fraction of prime numbers 1 above a multiple of 10 should approach 1 for all allowable residue classes.', 'duration': 26.53, 'max_score': 983.566, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ983566.jpg'}, {'end': 1111.237, 'src': 'embed', 'start': 1084.971, 'weight': 3, 'content': [{'end': 1090.096, 'text': "where n is any number and r is anything that's co-primed to n.", 'start': 1084.971, 'duration': 5.125}, {'end': 1094.039, 'text': "Remember, that means it doesn't share any factors with n bigger than 1.", 'start': 1090.096, 'duration': 3.943}, {'end': 1095.121, 'text': 'Instead of approaching 1.', 'start': 1094.039, 'duration': 1.082}, {'end': 1100.65, 'text': 'fourth, as x goes to infinity, that proportion goes to 1, divided by phi,', 'start': 1095.121, 'duration': 5.529}, {'end': 1106.239, 'text': 'where phi is that special function I mentioned earlier that gives the number of possible residues co-primed to n.', 'start': 1100.65, 'duration': 5.589}, {'end': 1111.237, 'text': 'And in case this is too clear for the reader, you might see it buried in more notation,', 'start': 1107.475, 'duration': 3.762}], 'summary': 'As x goes to infinity, the proportion goes to 1 divided by phi, where phi gives the number of possible residues co-prime to n.', 'duration': 26.266, 'max_score': 1084.971, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ1084971.jpg'}, {'end': 1149.405, 'src': 'embed', 'start': 1124.284, 'weight': 1, 'content': [{'end': 1129.348, 'text': "In some contexts, when people refer to Dirichlet's theorem, they refer to a much more modest statement,", 'start': 1124.284, 'duration': 5.064}, {'end': 1135.293, 'text': 'which is simply that each of these residue classes that might have infinitely many primes does have infinitely many.', 'start': 1129.348, 'duration': 5.945}, {'end': 1143, 'text': 'In order to prove this, what Dirichlet did was show that the primes are just as dense in any one of these residue classes as in any other.', 'start': 1136.234, 'duration': 6.766}, {'end': 1149.405, 'text': 'For example, imagine, someone asks you to prove that there are infinitely many primes ending in the number 1,', 'start': 1144.52, 'duration': 4.885}], 'summary': "Dirichlet's theorem states that each residue class with infinitely many primes also has infinitely many primes.", 'duration': 25.121, 'max_score': 1124.284, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ1124284.jpg'}, {'end': 1193.414, 'src': 'embed', 'start': 1162.935, 'weight': 6, 'content': [{'end': 1167.823, 'text': "Now the proof, well, it's way more involved than would be reasonable to show here.", 'start': 1162.935, 'duration': 4.888}, {'end': 1173.111, 'text': 'One interesting fact worth mentioning is that it relies heavily on complex analysis,', 'start': 1168.424, 'duration': 4.687}, {'end': 1177.678, 'text': 'which is the study of doing calculus with functions whose inputs and outputs are complex numbers.', 'start': 1173.111, 'duration': 4.567}, {'end': 1180.081, 'text': 'Now that might seem weird right?', 'start': 1178.439, 'duration': 1.642}, {'end': 1186.748, 'text': 'I mean prime numbers seem wholly unrelated to the continuous world of calculus, much less when complex numbers end up in the mix.', 'start': 1180.221, 'duration': 6.527}, {'end': 1193.414, 'text': 'But since the early 19th century, this is absolutely par for the course when it comes to understanding how primes are distributed.', 'start': 1187.228, 'duration': 6.186}], 'summary': 'Proof relies on complex analysis, linking prime numbers and calculus since 19th century.', 'duration': 30.479, 'max_score': 1162.935, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ1162935.jpg'}, {'end': 1221.344, 'src': 'embed', 'start': 1196.736, 'weight': 2, 'content': [{'end': 1202.538, 'text': 'Understanding the distribution of primes in residue classes like this continues to be relevant in modern research too.', 'start': 1196.736, 'duration': 5.802}, {'end': 1208.98, 'text': 'Some of the recent breakthroughs on small gaps between primes, edging towards that ever-elusive twin-prime conjecture,', 'start': 1202.998, 'duration': 5.982}, {'end': 1213.801, 'text': 'have their basis in understanding how primes split up among these kinds of residue classes.', 'start': 1208.98, 'duration': 4.821}, {'end': 1221.344, 'text': 'Okay, looking back over the puzzle, I want to emphasize something.', 'start': 1218.363, 'duration': 2.981}], 'summary': 'Research shows primes distribution in residue classes is relevant in modern research, influencing breakthroughs on small prime gaps.', 'duration': 24.608, 'max_score': 1196.736, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ1196736.jpg'}, {'end': 1297.213, 'src': 'embed', 'start': 1258.513, 'weight': 5, 'content': [{'end': 1264.217, 'text': "And this isn't a coincidence that a fairly random question like this can lead you to an important and deep fact for math.", 'start': 1258.513, 'duration': 5.704}, {'end': 1270.402, 'text': 'What it means for a piece of math to be important and deep is that it connects to many other topics.', 'start': 1264.778, 'duration': 5.624}, {'end': 1278.588, 'text': "So even an arbitrary exploration of numbers, as long as it's not too arbitrary, has a good chance of stumbling into something meaningful.", 'start': 1271.443, 'duration': 7.145}, {'end': 1286.067, 'text': "Sure, you'll get a much more concentrated dosage of important facts by going through a textbook or a course,", 'start': 1280.444, 'duration': 5.623}, {'end': 1288.248, 'text': 'and there will be many fewer uninteresting dead ends.', 'start': 1286.067, 'duration': 2.181}, {'end': 1291.99, 'text': 'But there is something special about rediscovering these topics on your own.', 'start': 1288.948, 'duration': 3.042}, {'end': 1297.213, 'text': "If you effectively reinvent Euler's totient function before you've ever seen it defined,", 'start': 1292.73, 'duration': 4.483}], 'summary': 'Exploring math can lead to discovering important and deep facts, even through arbitrary explorations of numbers.', 'duration': 38.7, 'max_score': 1258.513, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ1258513.jpg'}], 'start': 983.566, 'title': 'Prime number residue classes', 'summary': "Discusses the distribution of prime numbers in residue classes, as x approaches infinity, the fraction of prime numbers 1 above a multiple of 10 should approach 1/4, and likewise for other allowable residue classes like 3, 7, and 9. dirichlet's theorem states that each residue class with infinitely many primes also has infinitely many primes, and it relies heavily on complex analysis. the chapter also explores the value of independent exploration in mathematics.", 'chapters': [{'end': 1123.304, 'start': 983.566, 'title': 'Prime number residue classes', 'summary': 'Discusses the distribution of prime numbers in residue classes, stating that as x approaches infinity, the fraction of prime numbers 1 above a multiple of 10 should approach 1/4, and likewise for other allowable residue classes like 3, 7, and 9. the theorem, established by dirichlet in 1837, illustrates an even spread between different allowable residue classes and holds a significant place in modern analytic number theory.', 'duration': 139.738, 'highlights': ['The theorem states that as x approaches infinity, the fraction of prime numbers 1 above a multiple of 10 should approach 1/4, and likewise for other allowable residue classes like 3, 7, and 9.', 'The theorem established by Dirichlet in 1837 illustrates an even spread between the 20 different allowable residue classes, holding a significant place in modern analytic number theory.', 'The proportion of primes with a residue of r mod n, where n is any number and r is co-primed to n, goes to 1 divided by phi as x goes to infinity, where phi gives the number of possible residues co-primed to n.']}, {'end': 1221.344, 'start': 1124.284, 'title': "Dirichlet's theorem on prime distribution", 'summary': "Discusses dirichlet's theorem, which states that each residue class with infinitely many primes also has infinitely many primes, and it relies heavily on complex analysis, with recent breakthroughs in small gaps between primes being based on understanding how primes split up among residue classes.", 'duration': 97.06, 'highlights': ["Dirichlet's theorem states that each residue class with infinitely many primes also has infinitely many primes, and it relies heavily on complex analysis.", 'Understanding the distribution of primes in residue classes like this continues to be relevant in modern research, with recent breakthroughs on small gaps between primes being based on this understanding.', "The proof of Dirichlet's theorem relies heavily on complex analysis, which is the study of doing calculus with functions whose inputs and outputs are complex numbers."]}, {'end': 1316.999, 'start': 1222.051, 'title': 'Rediscovering math: exploring numbers', 'summary': 'Explores the value of independent exploration in mathematics, highlighting the potential for significant discoveries and a deeper understanding of mathematical concepts through self-discovery and play.', 'duration': 94.948, 'highlights': ["Engaging in independent exploration in mathematics can lead to significant discoveries and a deeper understanding of mathematical concepts, potentially connecting to important theorems such as Dirichlet's theorem.", 'Rediscovering mathematical concepts independently can lead to a more effective learning process and foster a deeper familiarity with the material.', "Exploring numbers independently can lead to meaningful discoveries and a deeper understanding of mathematical concepts, potentially connecting to important theorems such as Dirichlet's theorem."]}], 'duration': 333.433, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/EK32jo7i5LQ/pics/EK32jo7i5LQ983566.jpg', 'highlights': ['The theorem states that as x approaches infinity, the fraction of prime numbers 1 above a multiple of 10 should approach 1/4, and likewise for other allowable residue classes like 3, 7, and 9.', "Dirichlet's theorem states that each residue class with infinitely many primes also has infinitely many primes, and it relies heavily on complex analysis.", 'Understanding the distribution of primes in residue classes like this continues to be relevant in modern research, with recent breakthroughs on small gaps between primes being based on this understanding.', 'The proportion of primes with a residue of r mod n, where n is any number and r is co-primed to n, goes to 1 divided by phi as x goes to infinity, where phi gives the number of possible residues co-primed to n.', 'The theorem established by Dirichlet in 1837 illustrates an even spread between the 20 different allowable residue classes, holding a significant place in modern analytic number theory.', "Engaging in independent exploration in mathematics can lead to significant discoveries and a deeper understanding of mathematical concepts, potentially connecting to important theorems such as Dirichlet's theorem.", "The proof of Dirichlet's theorem relies heavily on complex analysis, which is the study of doing calculus with functions whose inputs and outputs are complex numbers.", 'Rediscovering mathematical concepts independently can lead to a more effective learning process and foster a deeper familiarity with the material.', "Exploring numbers independently can lead to meaningful discoveries and a deeper understanding of mathematical concepts, potentially connecting to important theorems such as Dirichlet's theorem."]}], 'highlights': ['The theorem states that as x approaches infinity, the fraction of prime numbers 1 above a multiple of 10 should approach 1/4, and likewise for other allowable residue classes like 3, 7, and 9.', "Dirichlet's theorem states that each residue class with infinitely many primes also has infinitely many primes, and it relies heavily on complex analysis.", 'The concept of residue classes mod 44 illustrates the distribution of prime numbers and co-prime numbers within the spiral diagram, demonstrating the relationship between prime numbers and residue classes.', 'The proportion of primes with a residue of r mod n, where n is any number and r is co-primed to n, goes to 1 divided by phi as x goes to infinity, where phi gives the number of possible residues co-primed to n.', 'The explanation of how prime numbers form gaps in the visualization due to their relation with the residue classes of 44, highlighting the concept of residue classes and its impact on the distribution of prime numbers.', "The emergence of Archimedean spirals and patterns when zooming out from prime numbers, revealing 20 spirals and 280 rays, leads to the important theorem of Dirichlet's theorem in number theory.", 'The significance of 710 radians being extremely close to a whole number of full turns, visually represented by the point almost exactly on the x-axis and analytically as 710 divided by 2 pi rotations being 113.000095, implying that the angle of each new point is almost exactly the same as the last one when moving forward by steps of 710.', 'The rational approximation of 2 pi as 44 sevenths, leading to the clean separation of residue classes mod 44.', 'The pattern involves plotting points in polar coordinates where both coordinates are prime numbers.', 'The chapter emphasizes that there is no practical reason for this exploration, as it is purely for fun and data visualization.']}