Why do prime numbers make these spirals?

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<div class="cue style-scope ytd-transcript-body-renderer">the full title of this video might be</div>

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<div class="cue style-scope ytd-transcript-body-renderer">something like how pretty but pointless</div>

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<div class="cue style-scope ytd-transcript-body-renderer active">patterns in polar plots of primes prompt</div>

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<div class="cue style-scope ytd-transcript-body-renderer">pretty important ponderings on</div>

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<div class="cue style-scope ytd-transcript-body-renderer">properties of those primes I first saw</div>

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<div class="cue style-scope ytd-transcript-body-renderer">this pattern that I'm about to show you</div>

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<div class="cue style-scope ytd-transcript-body-renderer">in a question on the math Stack Exchange</div>

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<div class="cue style-scope ytd-transcript-body-renderer">it was asked by a user under the named</div>

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<div class="cue style-scope ytd-transcript-body-renderer">why mark and answered by Greg Martin and</div>

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<div class="cue style-scope ytd-transcript-body-renderer">it relates to the distribution of prime</div>

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<div class="cue style-scope ytd-transcript-body-renderer">numbers together with rational</div>

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<div class="cue style-scope ytd-transcript-body-renderer">approximations for pi you see what the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">user had been doing was playing around</div>

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<div class="cue style-scope ytd-transcript-body-renderer">with data in polar coordinates as a</div>

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<div class="cue style-scope ytd-transcript-body-renderer">quick reminder so we're all in the same</div>

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<div class="cue style-scope ytd-transcript-body-renderer">this means labeling points in 2d space</div>

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<div class="cue style-scope ytd-transcript-body-renderer">not with the usual XY coordinates but</div>

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<div class="cue style-scope ytd-transcript-body-renderer">instead with a distance from the origin</div>

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<div class="cue style-scope ytd-transcript-body-renderer">commonly called R for radius together</div>

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<div class="cue style-scope ytd-transcript-body-renderer">with the angle that that radial line</div>

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<div class="cue style-scope ytd-transcript-body-renderer">makes with the horizontal commonly</div>

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<div class="cue style-scope ytd-transcript-body-renderer">called theta and for our purposes this</div>

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<div class="cue style-scope ytd-transcript-body-renderer">angle will be measured in radians which</div>

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<div class="cue style-scope ytd-transcript-body-renderer">basically means that an angle of pi is</div>

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<div class="cue style-scope ytd-transcript-body-renderer">halfway around and then 2 pi is a full</div>

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<div class="cue style-scope ytd-transcript-body-renderer">circle and notice polar coordinates are</div>

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<div class="cue style-scope ytd-transcript-body-renderer">not unique in the sense that adding 2 pi</div>

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<div class="cue style-scope ytd-transcript-body-renderer">to that second coordinate doesn't change</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the location that this pair of numbers</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is referring to the pattern that we'll</div>

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<div class="cue style-scope ytd-transcript-body-renderer">look at centers around plotting points</div>

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<div class="cue style-scope ytd-transcript-body-renderer">where both of these coordinates are a</div>

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<div class="cue style-scope ytd-transcript-body-renderer">given prime number there is no practical</div>

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<div class="cue style-scope ytd-transcript-body-renderer">reason to do this it's purely fun we're</div>

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<div class="cue style-scope ytd-transcript-body-renderer">just frolicking around in the playground</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of data visualization and to get a sense</div>

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<div class="cue style-scope ytd-transcript-body-renderer">for what it means look at all the whole</div>

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<div class="cue style-scope ytd-transcript-body-renderer">numbers rather than just the primes the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">point 1 1 sits a distance one away from</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the origin with an angle of 1 Radian</div>

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<div class="cue style-scope ytd-transcript-body-renderer">which actually means this arc is the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">same length as that radial line and then</div>

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<div class="cue style-scope ytd-transcript-body-renderer">2 2 has twice that angle and twice the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">distance and to get to 3 3 you rotate</div>

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<div class="cue style-scope ytd-transcript-body-renderer">one more Radian with a total angle</div>

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<div class="cue style-scope ytd-transcript-body-renderer">that's now slightly less than a half</div>

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<div class="cue style-scope ytd-transcript-body-renderer">turn since 3 is slightly less than pi</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and you step one unit farther away from</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the origin I really want you to make</div>

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<div class="cue style-scope ytd-transcript-body-renderer">sure that it's clear what's being</div>

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<div class="cue style-scope ytd-transcript-body-renderer">plotted because everything that follows</div>

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<div class="cue style-scope ytd-transcript-body-renderer">depends on understanding it each step</div>

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<div class="cue style-scope ytd-transcript-body-renderer">forward is like the tip of a clock hand</div>

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<div class="cue style-scope ytd-transcript-body-renderer">which rotates one Radian with each tick</div>

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<div class="cue style-scope ytd-transcript-body-renderer">a little less than a sixth of a turn and</div>

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<div class="cue style-scope ytd-transcript-body-renderer">it grows by one unit at each step as you</div>

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<div class="cue style-scope ytd-transcript-body-renderer">continue these points spiral outwards</div>

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<div class="cue style-scope ytd-transcript-body-renderer">forming what's known in the business as</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Archimedean spiral now if you make the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">admittedly arbitrary move to knock out</div>

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<div class="cue style-scope ytd-transcript-body-renderer">everything except the prime numbers it</div>

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<div class="cue style-scope ytd-transcript-body-renderer">initially looks quite random</div>

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<div class="cue style-scope ytd-transcript-body-renderer">after all primes are famous for their</div>

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<div class="cue style-scope ytd-transcript-body-renderer">chaotic and difficult to predict</div>

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<div class="cue style-scope ytd-transcript-body-renderer">behavior but when you zoom out what you</div>

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<div class="cue style-scope ytd-transcript-body-renderer">start to see are these very clear</div>

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<div class="cue style-scope ytd-transcript-body-renderer">galactic seeming spirals and what's</div>

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<div class="cue style-scope ytd-transcript-body-renderer">weird is some of the arms seem to be</div>

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<div class="cue style-scope ytd-transcript-body-renderer">missing then zooming out even further</div>

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<div class="cue style-scope ytd-transcript-body-renderer">those spirals give way to a different</div>

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<div class="cue style-scope ytd-transcript-body-renderer">pattern</div>

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<div class="cue style-scope ytd-transcript-body-renderer">[Music]</div>

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<div class="cue style-scope ytd-transcript-body-renderer">these many different outward pointing</div>

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<div class="cue style-scope ytd-transcript-body-renderer">rays and those rays seem to mostly come</div>

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<div class="cue style-scope ytd-transcript-body-renderer">in clumps of four but there's the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">occasional gap like a comb missing its</div>

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<div class="cue style-scope ytd-transcript-body-renderer">teeth</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the question for you and me naturally is</div>

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<div class="cue style-scope ytd-transcript-body-renderer">what on earth is going on here where do</div>

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<div class="cue style-scope ytd-transcript-body-renderer">these spirals come from and why do we</div>

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<div class="cue style-scope ytd-transcript-body-renderer">instead get straight lines at this</div>

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<div class="cue style-scope ytd-transcript-body-renderer">larger scale if you wanted you could ask</div>

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<div class="cue style-scope ytd-transcript-body-renderer">a more quantitative question and count</div>

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<div class="cue style-scope ytd-transcript-body-renderer">that there are 20 total spirals</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and then up at that larger scale if you</div>

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<div class="cue style-scope ytd-transcript-body-renderer">patiently went through each ray you'd</div>

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<div class="cue style-scope ytd-transcript-body-renderer">count up a total of 280 and so this adds</div>

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<div class="cue style-scope ytd-transcript-body-renderer">a further mystery of where these numbers</div>

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<div class="cue style-scope ytd-transcript-body-renderer">are coming from and why they would arise</div>

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<div class="cue style-scope ytd-transcript-body-renderer">from primes now this is shocking and</div>

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<div class="cue style-scope ytd-transcript-body-renderer">beautiful and you might think that it</div>

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<div class="cue style-scope ytd-transcript-body-renderer">suggests some divine hidden symmetry</div>

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<div class="cue style-scope ytd-transcript-body-renderer">within the primes but to steady your</div>

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<div class="cue style-scope ytd-transcript-body-renderer">expectations I should say that the fact</div>

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<div class="cue style-scope ytd-transcript-body-renderer">that the person asking this question on</div>

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<div class="cue style-scope ytd-transcript-body-renderer">math exchange jumped right into prime</div>

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<div class="cue style-scope ytd-transcript-body-renderer">numbers makes the puzzle a little</div>

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<div class="cue style-scope ytd-transcript-body-renderer">misleading if you look at all of the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">whole numbers not just the primes as you</div>

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<div class="cue style-scope ytd-transcript-body-renderer">zoom out you see very similar spirals</div>

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<div class="cue style-scope ytd-transcript-body-renderer">they're much cleaner and now there's 44</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of them instead of 20 but it means that</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the question of where the spirals come</div>

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<div class="cue style-scope ytd-transcript-body-renderer">from is perhaps disappointingly</div>

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<div class="cue style-scope ytd-transcript-body-renderer">completely separate from the question of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">what happens when we limit our view to</div>

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<div class="cue style-scope ytd-transcript-body-renderer">primes but don't be too disappointed</div>

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<div class="cue style-scope ytd-transcript-body-renderer">because both these questions are still</div>

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<div class="cue style-scope ytd-transcript-body-renderer">phenomenal puzzles there's a very</div>

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<div class="cue style-scope ytd-transcript-body-renderer">satisfying answer for the spirals and</div>

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<div class="cue style-scope ytd-transcript-body-renderer">even if the primes don't cause the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">spirals asking what goes on when you</div>

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<div class="cue style-scope ytd-transcript-body-renderer">filter for those Prime's does lead you</div>

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<div class="cue style-scope ytd-transcript-body-renderer">to one of the most important theorems</div>

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<div class="cue style-scope ytd-transcript-body-renderer">about the distribution of prime numbers</div>

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<div class="cue style-scope ytd-transcript-body-renderer">known in number theory as derice lace</div>

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<div class="cue style-scope ytd-transcript-body-renderer">theorem to kick things off let's zoom</div>

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<div class="cue style-scope ytd-transcript-body-renderer">back in a little bit smaller did you</div>

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<div class="cue style-scope ytd-transcript-body-renderer">notice that as we were zooming out there</div>

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<div class="cue style-scope ytd-transcript-body-renderer">were these six little spirals this</div>

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<div class="cue style-scope ytd-transcript-body-renderer">offers a good starting point to explain</div>

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<div class="cue style-scope ytd-transcript-body-renderer">what's happening in the two larger</div>

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<div class="cue style-scope ytd-transcript-body-renderer">patterns notice how all the multiples of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">six form one arm of this spiral</div>

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<div class="cue style-scope ytd-transcript-body-renderer">then the next one is every integer</div>

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<div class="cue style-scope ytd-transcript-body-renderer">that's one above a multiple of six then</div>

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<div class="cue style-scope ytd-transcript-body-renderer">after that and includes all the numbers</div>

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<div class="cue style-scope ytd-transcript-body-renderer">two above a multiple of six and so on</div>

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<div class="cue style-scope ytd-transcript-body-renderer">why is that well remember that each step</div>

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<div class="cue style-scope ytd-transcript-body-renderer">forward in the sequence involves a turn</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of one Radian so when you count up by</div>

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<div class="cue style-scope ytd-transcript-body-renderer">six you've turned a total of six radians</div>

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<div class="cue style-scope ytd-transcript-body-renderer">which is a little bit less than two pi a</div>

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<div class="cue style-scope ytd-transcript-body-renderer">full turn so every time you count up by</div>

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<div class="cue style-scope ytd-transcript-body-renderer">six you've almost made a full turn it's</div>

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<div class="cue style-scope ytd-transcript-body-renderer">just a little less another six steps a</div>

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<div class="cue style-scope ytd-transcript-body-renderer">slightly smaller angle six more steps</div>

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<div class="cue style-scope ytd-transcript-body-renderer">smaller still and so on with this angle</div>

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<div class="cue style-scope ytd-transcript-body-renderer">changing gently enough that it gives the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">illusion of a single curving line when</div>

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<div class="cue style-scope ytd-transcript-body-renderer">you limit the view to prime numbers all</div>

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<div class="cue style-scope ytd-transcript-body-renderer">but two of these spiral arms they'll go</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and think about it a prime number can't</div>

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<div class="cue style-scope ytd-transcript-body-renderer">be a multiple of six and it also can't</div>

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<div class="cue style-scope ytd-transcript-body-renderer">be two above a multiple of six unless</div>

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<div class="cue style-scope ytd-transcript-body-renderer">it's two or four above a multiple of six</div>

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<div class="cue style-scope ytd-transcript-body-renderer">since all of those are even numbers it</div>

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<div class="cue style-scope ytd-transcript-body-renderer">also can't be three above a multiple of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">six unless it's the number three itself</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">since all of those are divisible by</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">three so at least of the smaller-scale</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">nothing magical is going on and while</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">we're in the simpler context let me</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">introduce some terminology that</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">mathematicians use each one of these</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">sequences where you're counting up by 6</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">is fancifully called a residue class mod</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">6 the word residue here is sort of an</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">overdramatic way of saying remainder and</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">mod means something like where the thing</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">you divide by is so for example six goes</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">into twenty three times and it leaves a</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">remainder of two so 20 has a residue of</div>

</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">together with all the other numbers</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">leaving a remainder of two when the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">thing you divide by is six you have a</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">full residue class mod six I know that</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that sounds like the world's most</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">pretentious way of saying everything to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">above a multiple of six but this is the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">standard jargon and it is actually handy</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">to have some words for the idea so</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">looking at our diagram in the lingo</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">each of these spiral arms corresponds to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">a residue class mod six and the reason</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">we see them is that six is close to two</div>

</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">turning six radians is almost a full</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">turn and the reason we see only two of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">them when filtering for primes is that</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">all prime numbers are either one or five</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">above a multiple of six with the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">exceptions of two and three with all</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that as a warm-up let's think about the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">larger scale in the same way that six</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">steps is close to a full turn taking</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">forty-four steps is very close to a</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">whole number of turns let's compute it</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">there are two pi radians per rotation</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">right so taking forty four steps turning</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">forty four radians gives a total of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">forty four divided by two pi rotations</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">which comes out to be just barely above</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">seven full turns you could also write</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">this by saying that forty four sevenths</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">is a close approximation for two pi</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">which some of you may better recognize</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">as the famous 22 sevenths approximation</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">for pi what this means is when you count</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">up by multiples of 44 in the diagram</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">each point has almost the same angle as</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the last one just a little bit bigger so</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">as you continue on with more and more we</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">get this very gentle spiral as the angle</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">increases very slowly similarly all the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">numbers one above a multiple of 44 make</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">another spiral but rotated one Radian</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">counterclockwise same for everything to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">above a multiple of 44 and so on</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">eventually filling out the full diagram</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">to phrase it with our fancier language</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">each of these spiral arms shows a</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">residue class mod 44 and maybe now you</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">can tell me what happens when we limit</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">our view to prime numbers Prime's</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">cannot be a multiple of 44 so that arm</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">won't be visible nor can a prime be two</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">above a multiple of 44 or four above and</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">so on since all those residue classes</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">have nothing but even numbers likewise</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">any multiples of eleven can't be prime</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">except for 11 itself so the spiral of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">numbers 11 above of multiple of 44 they</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">won't be visible and neither will the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">spiral of numbers 33 above a multiple of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">44 this is what gives the picture those</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Milky Way seeming gaps each spiral were</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">left with is a residue class that</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">doesn't share any prime factors with 44</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and within each one of those arms that</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">we can't reject out of hand the prime</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">numbers seem to be sort of randomly</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">distributed and that's a fact that I'd</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">like you to tuck away we'll return to it</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">later this is another good chance to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">inject some of the jargon that</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">mathematicians use what we care about</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">right here are all the numbers between 0</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">& 43 that don't share a prime factor</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">with 44 right the ones that aren't even</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and they also aren't divisible by 11</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">when two numbers don't share any factors</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">like this we call them relatively prime</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">or also Co prime in this example you</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">could count that there are 20 different</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">numbers between 1 and 44 that are</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">co-prime to 44 and this is a fact that a</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">number theorist would compactly right by</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">saying fie of 44 equals 20 where the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Greek letter fee here refers to euler's</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">totient function yet another needlessly</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">fancy word which is defined to be the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">number of integers from 1 up to n which</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">are co-prime to n it comes up enough</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that it's handy to have compact notation</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">more obscurely and I had never heard</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">this before but I find it too delightful</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">not to tell these numbers are sometimes</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">called the tota tips of n back to the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">main thread in short what the user on</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">math exchange was seeing are two</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">unrelated pieces of number theory but</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">illustrated in one drawing the first is</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that 44 sevenths is a very close</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">rational approximation</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">for 2pi which results in the residue</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">classes mod 44 being cleanly separated</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">out the second is that many of these</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">residue classes contain zero prime</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">numbers or sometimes just one so they</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">won't show up but on the other hand</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Prime's do show up plentifully enough in</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">all 20 of the other residue classes that</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">it makes these spiral arms visible and</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">at this point maybe you can predict</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">what's going on at the larger scale just</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">as 6 radians is vaguely close to a full</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">turn and 44 radians is quite close to 7</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">full turns it just so happens that 710</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">radians is extremely close to a whole</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">number of full turns visually you can</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">see this by the fact that the point ends</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">up almost exactly on the x axis but it's</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">more compelling analytically 710 radians</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">is 710 divided by 2 pi rotations which</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">works out to be 113 point 0 0 0 0 0 9 5</div>

</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">another form it's saying the 710 113 is</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">a close approximation for 2pi which is</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">more commonly seen in saying that 355</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">over 113 is a very good approximation</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">for pi now if you want to understand</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">where these rational approximations are</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">coming from and what it means for one</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">like this to be unusually good like way</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">better than you would get for fee or e</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">or square root of 2 or other famous</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Irrational's I highly recommend taking a</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">look at this great mythology video for</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">our storyline though it means that when</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">you move forward by steps of 710 the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">angle of each new point is almost</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">exactly the same as the last one just</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">microscopically bigger even very far out</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">one of these sequences looks like a</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">straight line and of course the other</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">residue class is mod 710 also formed</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">these nearly straight lines 710 is a big</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">number though so when all of them are</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">on-screen and there's only so many</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">pixels on the screen it's a little hard</div>

</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">so in this case it's actually easier to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">see when we limit the view to primes</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">where you don't see many of those</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">residue classes in reality with a little</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">further zooming you can see that there</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">actually is a very gentle spiral to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">these but the fact that it takes so long</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">to become prominent is a wonderful</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">illustration maybe the best illustration</div>

</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">approximation this is for 2pi tying up</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the remaining loose thread here if you</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">want to understand what happens when you</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">filter for primes it's entirely</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">analogous to what we did before the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">factors of 710 are seventy one five and</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">two so if the remainder or residue is</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">divisible by any of those then so is the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">number when you pull up all of the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">residue classes with odd numbers it</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">looks like every other array in the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">otherwise quite crowded picture and then</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">of those that remain these are the ones</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that are divisible by five which are</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">nice and evenly spaced at every fifth</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">line notice the fact the prime numbers</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">never show up in any of these is what</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">explains the pattern of the lines we saw</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">at the beginning coming in clumps of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">four and moreover of those remaining</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">these four residue classes are the ones</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that are divisible by 71 so the primes</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">aren't going to show up there and that's</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">what explains why the clumps of four</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that we saw occasionally have like a</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">missing tooth in your cone and if you</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">were wondering where that number 280</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">came from it comes from counting how</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">many of the numbers from 1 up to 710</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">don't share any prime factors with 710</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">these are the ones that we can't rule</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">out for including Prime's based on some</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">obvious divisibility consideration this</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">of course doesn't guarantee that any</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">particular one will contain prime</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">numbers but at least empirically when</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">you look at this picture it actually</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">seems like the primes are pretty evenly</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">distributed among the remaining classes</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">wouldn't you agree</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">this last point is actually the most</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">interesting observation of the whole</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">deal it relates to a pretty deep fact in</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">number theory known as derice lace</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">theorem to take a simpler example than</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">residue classes mod 7/10 think of those</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">mod 10 because we write numbers in base</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">10 this is the same thing as grouping</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">numbers together by what their last</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">digit is everything whose last digit is</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">0 is a residue class everything whose</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">last to do this one is another residue</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">class and so on other than two prime</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">numbers can't have an even number as</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">their last digit since that means</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">they're even and likewise any prime</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">bigger than 5 can't in and a 5 there's</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">nothing surprising there that's one of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the first facts you observe when you</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">learn about prime numbers anything</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">bigger than 5 has to end in either a 1 a</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">3 a 7 or a 9 a much more nuanced</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">question though is how exactly these</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Prime's are divvied up among those</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">remaining 4 groups here let's make a</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">quick histogram counting through each</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">prime number where the bars are going to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">show what proportion of the Prime's that</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">we've seen so far have a given last</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">digit so in particular the 2 and the 5</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">slots should go down to 0 over time what</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">would you predict is gonna happen as we</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">move through more and more Prime's</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">well as we get a lot of them it seems</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">like a pretty even spread between these</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">four classes about 25% each and probably</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that's what you would expect after all</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">why would prime numbers show some sort</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">of preference for one last digit over</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">another but Prime's aren't random they</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">are a definite sequence and they show</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">patterns in other ways and it's highly</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">non-obvious how you would prove</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">something like this or for that matter</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">how do you rigorously phrase what it is</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that you want to prove a mathematician</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">might go about it something like this if</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">you look at all the prime numbers less</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">than some big number X and you consider</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">what fraction of them are say one above</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">a multiple of ten that fraction should</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">approach one-fourth as X approaches</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">infinity and likewise for all of the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">other allowable residue classes like 3 &</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">7 & 9 of course there's nothing special</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">about 10 a similar fact should hold for</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">any other number considering our old</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">friends the residue classes menthe 44</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">for example let's make a similar</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">histogram showing what proportion of the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">primes show up in each one of these</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">[Music]</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">again as time goes on we see an even</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">spread between the 20 different</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">allowable residue classes which you can</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">think of in terms of each spiral arm</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">from our diagram having about the same</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">number of primes as each of the others</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">maybe that's what you would expect but</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">this is a shockingly hard fact to prove</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the first man who cracked this puzzle</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">was Dury Schley in 1837 and it forms one</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">of the crowning jewels at the foundation</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">of modern analytic number theory</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">histograms like these ones give a pretty</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">good illustration of what the theorem is</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">actually saying nevertheless you might</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">find it enlightening to see how it might</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">be written in a math text with all the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">fancy jargon and everything it's</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">essentially what we just saw 4/10 but</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">more general again you look at all of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the primes up to some bound X but</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">instead of asking for what proportion of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">them have a residue of say 1 mod 10 you</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">ask what proportion have a residue of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">our mod n where n is any number and R is</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">anything that's co-prime to n remember</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that means it doesn't share any factors</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">with n bigger than 1 instead of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">approaching 1/4 as X goes to infinity</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that proportion goes to 1 divided by V</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">of n where V is that special function I</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">mentioned earlier that gives the number</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">of possible residues co-prime to n and</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">in case this is too clear for the reader</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">you might see a buried in more notation</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">where this denominator and the numerator</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">are both written with a special prime</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">counting function the convention rather</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">confusingly is to use the symbol pi for</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">this function even though it's totally</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">unrelated to the number pi in some</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">contexts when people refer to to reach</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">lace theorem they refer to a much more</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">modest statement which is simply that</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">each of these residue classes that might</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">have infinitely many Prime's does have</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">infinitely many in order to prove this</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">what to really did was show that the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">primes are just as dense in any one of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">these residue classes as in any other</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">for example imagine someone asked you to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">prove that there are infinitely many</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">Prime's ending in the number 1 and the</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">way you do it is by showing that a</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">quarter of all the primes and</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">one together with the fact that there</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">are infinitely many primes which we've</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">known since Euclid this gives a stronger</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">statement and a much more interesting</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">one now the proof well it's way more</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">involved than would be reasonable to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">show here one interesting fact worth</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">mentioning is that it relies heavily on</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">complex analysis which is the study of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">doing calculus with functions whose</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">inputs and outputs are complex numbers</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">now that might seem weird right</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">I mean prime numbers seem wholly</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">unrelated to the continuous world of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">calculus much less when complex numbers</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">end up in the mix but since the early</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">19th century</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">this is absolutely par for the course</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">when it comes to understanding how</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">primes are distributed and this isn't</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">just antiquated technology either</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">understanding the distribution of primes</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">in residue classes like this continues</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">to be relevant in modern research to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">some of the recent breakthroughs on</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">small gaps between primes edging towards</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that ever elusive twin prime conjecture</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">have their basis in understanding how</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">prime split up among these kinds of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">residue classes okay looking back over</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the puzzle I want to emphasize something</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the original bit of data visualization</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">whimsy that led to these patterns</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">well like it doesn't matter no one cares</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">there's nothing special about plotting P</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">comma P in polar coordinates and most of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the initial mistery in these spirals</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">resulted from the artifacts that come</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">from dealing with integer number of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">radians which is kind of weird but on</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">the other hand this kind of play is</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">clearly worth it if the end result is a</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">line of questions that leads you to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">something like de richelieu theorem</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">which is important especially if it</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">inspires you to learn enough to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">understand the tactics of the underlying</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">proof no small task by the way</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and this isn't a coincidence that a</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">fairly random question like this can</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">lead you to an important and deep fact</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">for math what it means for a piece of</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">math to be important and deep is that it</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">connects to many other topics so even an</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">arbitrary exploration of numbers as long</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">as it's not too arbitrary has a good</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">chance of stumbling into something</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">meaningful sure you'll get a much more</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">concentrated dosage of important facts</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">by going through a textbook or a course</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">and there will be many fewer</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">uninteresting dead ends but there is</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">something special about rediscovering</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">these topics on your own if you</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">effectively reinvent boilers totient</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">function before you've ever seen it</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">defined or if you start wondering about</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">rational approximations before learning</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">about continued fractions or if you</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">seriously explore how Prime's are</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">divvied up between residue classes</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">before you've even heard the name to</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">reach lay that when you do learn those</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">topics you'll see them as familiar</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">friends not as arbitrary definitions and</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">that will almost certainly mean that you</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">learn it more effectively</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">[Music]</div>

</div>

<div class="cue style-scope ytd-transcript-body-renderer">[Music]</div>

</div>

</div>

<div id="label-text" class="style-scope yt-dropdown-menu">English (auto-generated)</div>

</div>

</div>

</div>

Why do prime numbers make these spirals?

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