Prime Pyramid (with 3Blue1Brown) - Numberphile

Grant Sanderson (from 3Blue1Brown) shows us a pyramid that spits out prime numbers - and then we dig deeper.
More links & stuff in full description below ↓↓↓
See all three videos in this series - Grant's Prime Pattern Trilogy: bit.ly/PrimePatternTrilogy
Grant's own false pattern video: • Researchers thought th...
Grant's channel is 3Blue1Brown: / 3blue1brown
More Grant on Numberphile: bit.ly/Grant_Numberphile
Grant on the Numberphile Podcast: • The Hope Diamond (with...
Numberphile is supported by the Simons Laufer Mathematical Sciences Institute (formerly MSRI): bit.ly/MSRINumberphile
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. www.simonsfoundation.org/outr...
And support from The Akamai Foundation - dedicated to encouraging the next generation of technology innovators and equitable access to STEM education - www.akamai.com/company/corpor...
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Special thanks to our friend Jeff for the accommodation and filming space.

[@numberphile]

Part 1 of this three-part interview is at: kzbin.info/www/bejne/oJmyk3-KZrGlnNE Part 3 of this three-part interview: STILL BEING EDITED

[@FebruaryHas30Days]

First reply

[@Anonymous-df8it]

Second reply

[@volodyadykun6490]

Still

[@Anonymous-df8it]

@@volodyadykun6490 Fourth reply

[@Jono4174]

Skewes’s number-th reply

[@tommihommi1]

Suddenly out of nowhere, a Function named after Euler appears. Feel like that's a fundamental rule of mathematics

[@zmaj12321]

Euler's totient function is REALLY essential to anything involving number theory. Not surprising.

[@tyle.s9084]

@Paolo Verri And Gauss always found out about it when he was four years old

[@otonanoC]

Everything in math was invented by Euler or Riemann.

[@louisrobitaille5810]

@@otonanoC Euler or Gauss* 😝. Riemann just built a few things on Gauss' work 👀.

[@tommihommi1]

@@zmaj12321 I only knew it as doing some neat thing for RSA.

[@volodyadykun6490]

"Prime numbers are, like, the sexiest numbers available" Grant Sanderson, 2022

[@1224chrisng]

as James Grime would point out, we do have Sexy Primes, twin primes with a gap of 6

[@birdbeakbeardneck3617]

Shheeeeeshh

[@lonestarr1490]

@@1224chrisng Dude! There might be children reading this thread!

[@dyld921]

Grant Sanderson is, like, the sexiest mathematician available.

[@kadefringe]

I phap on prime numbers indeed

[@Rubrickety]

That silently-corrected "1/3" at 3:38 may be the first error I've ever seen Grant make 😂. The man is as smooth as an infinitely-differentiable function.

[@theadamabrams]

For anyone confused, the correction 1/3 → 2/3 happens around 3:49

[@berryzhang7263]

Omg yeah I was so confused when I saw the error lol

[@leftaroundabout]

If he didn't make any errors _at all_ he would be smooth like an analytic function. But that would be boring, because then you could represent him entirely by his Taylor expansion. _Countably_ many values, that can't be enough!

[@axp_bubbles]

If you watch the live streams he did during early pandemic days he makes a lot of errors while writing, and is very candid about them. Just a genuinely humble and brilliant human being.

[@SmileyMPV]

@@leftaroundabout not all smooth functions are analytic though but any continuous function is still determined by its rational evaluations, so in order to not be determined by only countably many values you do need to be discontinuous :P

[@mxlexrd]

An unlisted video from an unlisted video? Now we're in a super exclusive club!

[@krokorok_]

:D

[@viliml2763]

What video did you come from? I came from a listed video.

[@mxlexrd]

@@viliml2763 It wasn't listed when I made the comment

[@ophello]

The first video wasn’t unlisted.

[@themathhatter5290]

@@ophello It was when Grant linked it in his own video

[@davidgillies620]

The length of successive Farey sequences is OEIS A005728. The Euler totient function is one of the foundational objects of number theory. The fact that the sequence here is one plus the sum of the first n values of the totient function is another of those neat links that almost feel numerological in nature. If memory serves, there have already been Numberphile videos on the link between the Stern-Brocot tree and Farey sequences on the one hand, and Farey sequences and Ford circles on the other.

[@juniperlovelace5262]

Its a special talent to make your thumbnails consistently look like something out of the 90s

[@conanichigawa]

Grant's explanation is awesome, but Brady's analogies make it more accessible to everyone.

[@deadlyshizzno]

Is the third video ever coming? Have been checking back since this one first dropped

[@ShenghuiYang]

Amazing connection between Euler totient function, Farey and mobius inversion in such a short video.

[@wehpudicabok6598]

Grant: "1/5, 2/5 --" me: "red fifth, blue fifth"

[@ps.2]

Oh, what a lot of fifths there are!

[@redtaileddolphin1875]

Your original video on farey sums and ford circle packing is probably my favorite on this channel, and one of my favorite on all of the internet. To watch them suddenly come up in this video was truly a treat

[@jazermano]

Since I read your comment and got intrigued, I went and found the video, titled "Funny Fractions and Ford Circles." It's dated at being roughly 7 years old. But it is still has the same awesome Numberphile feel to it. Nice to see some things haven't changed.

[@redtaileddolphin1875]

@@jazermano aw thanks! it’s honestly asmr for me I love how he says “probably” and “pinkie”. 10/10 all math videos should also be asmr

[@hlvaneeden]

The sum of digits of that last sequence is not 33, it is 37, which is prime :) (if you count 10 as two digits).

[@scottabroughton]

But if you insert 10 11s, it comes to 57, which is composite.

[@gaborszucs2788]

​@@scottabroughton except that for example it's not 1+10, rather, 1+1, which is not 11, so you skip that, plus 10+1 at the end. 57-2x2 is 53 which happens to be a prime... Who'll volunteer for 12?

[@scottabroughton]

@@gaborszucs2788 Can you provide a visual?

[@razlotan7504]

It's like if you watch only 3b1b videos you would think everyone is as attractive as Grant

[@Michael-cg7yz]

7:14 So, we can define it as a function based on the Euler's totient function. one of the definitions of ETF is: phi(n) = sum from k=1 to n of gcd(k,n)*cos(2pi*k/n) then, the sequence would be defined as: 1 + phi(1) + phi(2) + phi(3).... or to rewrite: g(t) = ([sum from n = 1 to t of phi(n)] + 1) and, it still outputs primes even after the break omitted values denoted with ( ), erroneous with [ ] g(t): 2, 3, 5, 7, 11, 13, (17), 19, 23, 29, (31), [33], (37), (41), 43, 47, (53), 59, (61), [65], (67), (71), 73, (79), [81], (83), (89), 97, (101), 103 i mean yes, it breaks worse each time but the only erroneous values up to 100 are [33], [65] and [81]

[@lonestarr1490]

So all we need is a different imperfect prime sequence to use in conjunction with it, where it is guaranteed that the two of them never fail at the same time.

[@panadrame3928]

The question then is what is the proportion of non prime sums of φ(n) for n

[@Michael-cg7yz]

@@panadrame3928 you mean this g(x) or Euler's totient function? I'm fairly sure the first one is independent of primes, so sometimes it'll hit them, sometimes, and that being the larger amount it'll miss them

[@Vaaaaadim]

We're reaching levels of unlisted that shouldn't even be possible

[@viliml2763]

What video did you come from? I came from a listed video.

[@Vaaaaadim]

@@viliml2763 part 1 When 3B1B's vid came out today, it linked to part 1, which was unlisted at that time.

[@anoopramakrishna]

3 3 Blue 1 Brown Videos in 1 Day😁 Inception much?

[@MichaelJamesActually]

Funny how Grant can talk about a sequence of numbers that really doesn't have any sort of significance, and I still enjoy watching it.

[@AllHailZeppelin]

After realizing that the total number of DIGITS in the 10th row stays prime (37), I got hopeful that maybe the number of digits would keep the pattern even if the number of elements (numbers) doesn’t. But alas, at the 11th row the number of digits is 37+2*φ(11), or 57… 😕

[@highlewelt9471]

Grant is always such a delight

[@dkranda]

@9:47 excuse me but Tim “The Moth” Hein is absolutely an A lister!

[@toycobra12]

I thought it was the guy from the KFC logo 😂

[@ericpeterson6520]

Is part 3 still in the works?

[@joelkronqvist6089]

Seems like it was published today

[@ifroad33]

Great mathematician. Great KZbin content creator. Charismatic as heck. We all wish to be Grant I presume.

[@neil5280]

I check back every day for Part 3.

[@neil5280]

Monday was pretty chill.

[@neil5280]

I don't have the stamina for commenting any more, but I am checking daily. Look forward to Part 3 whenever it arrives.

[@neil5280]

Happy New Year! 🎉

[@Uranyus36]

probably the most fascinating prime pattern that tricks everybody the most is the approximating prime-counting function which leads to the birth of skewes number. even tho skewes number is an over-overestimate i guess the actually point where the prime-counting function changes its size comparison to the actual number of primes < n would still be something huge (like 10 to the power several hundreds?). this completely blasts through the regime of small numbers a mortal could interpret of, but yet at some point the relatively big boys still gonna break the pattern.

[@EebstertheGreat]

I hope part 3 won't be unlisted. If I don't get notified when it's uploaded, I'll probably never see it.

[@dhoyt902]

The second number in the rows of Pascal triangle(the counting numbers) will evenly go into every number in the row IFF the number is prime.

[@JamalanJuda]

My two favorite channels coming together.

[@jamesepace]

Oh darn, part 3 isn't up yet, which means I'm going to close this tab and forget to come back to see the exciting conclusion. :(

[@andrewharrison8436]

😃I bet you have already subscribed.

[@jamesepace]

@@andrewharrison8436 Yeah, but if it's unlisted it doesn't show up in the subscriptions list.

[@deadlyshizzno]

I have been coming back here like twice a day waiting for part 3 to be linked in the pinned comment or description! I'm excited for that vid, I could listen to Grant talk about math forever

[@deadlyshizzno]

I'm still checking!

[@deadlyshizzno]

Why is it still being edited 😭

[@deadlyshizzno]

D:

[@deadlyshizzno]

I suppose the third video in this series is somewhere in the backlog now

[@deadlyshizzno]

:(

[@fuuryuuSKK]

DEEPER INTO THE VAULT WE GO

[@OwlRTA]

ENHANCE

[@ekxo1126]

@@OwlRTA i just answered on a comment which was an answer to a comment of an unlisted video that I reached from another unlisted video

[@viliml2763]

​@@ekxo1126 What video did you come from? I came from a listed video.

[@kruksog]

Actually did research work on Farey sums and polynomials and so on. Wild to see some of it shared here. Feels like a fever dream seeing this presented 🙃

[@razer1024]

Best video in a long while 🎉❤

[@jacksonstarky8288]

And the third video is still being edited. But I needed to watch this again anyway. Grant's explanations are so clear and understandable that I keep expecting his channel to come out with a follow-up to his Riemann zeta function video proving the Riemann hypothesis.

[@nikhilkenvetil1594]

What is this, a crossover episode? ❤Great stuff as always!

[@happyelephant5384]

This collab is legendary

[@xanderalaniz2298]

It would be interesting to see how this works in other Bases. Following the totient function of 10, would it break down in a similar manner in duodecimal, or is it merely a trick of numbers merely being close to each other?

[@andrewharrison8436]

The totient function is independent of base. It depens on common factors (or lack of them) not on the representation of the number.

[@FirstLast-gw5mg]

Will the 3rd video be published on one of your channels, so that we'll see it?

[@danieluran9555]

This is an unexpected follow up to Dr. Bonahon's video... Great!!

[@SpySappingMyKeyboard]

When adding even numbers (because it's symmetric) to small odd numbers (after the first) it's hard not to hit a prime

[@TaranovskiAlex]

So... how many times more I have to refresh the page to see the link to the 3rd part? Are you testing if page refreshes contribute to the views number?

[@arandomdiamond2]

According to what you said about it being related to the number of fractions with a maximum denominator, this can compute primes! You just need to check how many numbers are added at each step and for step i, if i-1 numbers were added, then i is prime. I checked up to i=3000 too.

[@arandomdiamond2]

Not very efficient for calculating big primes though

[@TheEternalVortex42]

This is just checking the definition of the Euler totient function for primes, since φ(p) = p-1.

[@arandomdiamond2]

@@TheEternalVortex42 Yes, but I found it interesting since Grant said the "Prime Pyramid" didn't produce primes, and I've never seen primes calculated this way before so I just thought it was cool.

[@SuperYoonHo]

Awesome video!

[@a0z9]

In each row ,the most numerous number is the prime but if tie always choose the prime you Know from the previus rows.

[@Par_and_syv_lovers56]

awesome collab

[@TheFakeMackie]

3b1b is a phenom channel. Great collab.

[@johnkonrath1115]

Loving the trilogy!

[@backwashjoe7864]

I have a reminder set to look for the 4th / "Resurrections" video in 18 years.

[@EPMTUNES]

Grants always been a great math communicator!

[@bad_manbot]

it would be interesting to see the sequence of numbers that are primes that he pyramid skips, and see if they hold any patterns we can recognize

[@SgtSupaman]

Another comment did the output to just over 100. Here are the skipped primes they came up with: 17, 31, 37, 41, 53, 61, 67, 71, 79, 83, 89, 101

[@jurjenbos228]

@@SgtSupaman This is not in the OEIS. But the sequence of denominators of Farey sequences is: A006843, and the sequence of numbers of Farey fractions (prime or not) is A005728.

[@bad_manbot]

@@SgtSupaman nothing quite jumps off the page at me. though it is interesting the differences between the skipped primes from one to the next. 4, 6, 4, 12, 8, 6, 4, 8, 4, 6, 12 way less variability than I expected - though i have a suspicion that this is more due to the "6n+1, 6n-1" nature of primes than anything else. (also given how densely packed they are at the lower end of the number line, as mentioned in this video.)

[@addymant]

Will you upload the third video unlisted?

[@kingdomadventures]

In this series I saw something I never saw before--veins popping out of Grant's arms. Teach has been lifting!

[@hyftar]

Question about the prime pyramid, would the sequence still break if we used another base? (i.e. Would the same sequence in base 16, break at 16?)

[@MichaelDarrow-tr1mn]

it's not a base 10 specific thing

[@fidgettyspinner3028]

A nice mathematician's pause when that second "1/3" is noticed and fixed offscreen for the next section.

[@Sajatzsiraf]

This is super cool :) thank you for sharing this with us!

[@JamesJoyceJazz]

i want the third ep right now pls thanks in advance loving the material

[@toferg.8264]

4:22 so far it is a repeat of the Stern Brocot Sequence and the Funny Fractions video. Which is fine :) . I hope there is more.

[@deadlyshizzno]

Part 3 is finally out! Thanks for listening to the like 5 people that were asking for it in this comment section lol :D

[@bumbleandsimba]

NUMBERPHILE I LOVE YOU'RE VIDEOS 💗

[@lucas.cardoso]

If 1 was a prime number, then the first prime actor would be Sylvester StallONE.

[@ZacThompson]

3 brown paper videos: you should do 1 on blue paper with him just to complete the inversion

[@deadlyshizzno]

Guys the description changed from "STILL BEING EDITED" to "soon"

[@thatoneginger]

Grant is def a prime number, wish we’d see more of him on his home channel, but pie guy is cute too 😊

[@francescos7361]

Thanks .

[@AidanRatnage]

Suddenly, it's not unlisted anymore!

[@TheCapcarap]

This is the ultimate video

[@senthilkumaran5255]

neat sleight of hand at 3:47 :)

[@zerosir1852]

My three inventions able to change the all history of mathematics. (1) The Easy Number Theory (2) The Original Remainder Theorem (3) The Prime Pyramid Theorem

[@anonymoususer2756]

Thought this was going in the direction of the Stern-Brocot sequence at first

[@abuzzedwhaler7949]

Papa Grant here to give us some key geometric intuitions

[@RobertMStahl]

Rydberg is watching (finite structure) crystal eyes ...hydrino autopoiesis?

[@CorrectHorseBatteryStaple472]

7:10 Damn it, it's that Euler guy, again!

[@smizmar8]

The quip about 3b1b being "A list" haha, you certainly are too tho Bradey, I literally started learning math in my 20's because of your channels! :D

[@leobarlach]

That's the funny addition video! Classic!

[@bstlang]

On the line for number 10 is doesn't break if you count digits, since it becomes 37, not 33.

[@axelnilsson6478]

Poor Tim!

[@johnbumbledore]

What if you use a radix other than base ten. May be base 14 or base 22?

[@divisionzero715]

The function is irrespective of base, it shouldn't matter.

[@Mike-lx9qn]

2:10 unit divisors 2:54 4:10: okay, inbetween 2:17 7:00

[@kp2k]

super cool

[@kirkanos771]

It crashed at 10 but what if we count in base 16 and replace 10 by A. Its 1 less digit. Augmenting the base should delay the moment it crashes, is it ?

[@aceman0000099]

I also wondered if it fails at 10 because of base 10. It may be pure coincidence

[@user-qo3qm7ud1d]

It does not depend on base of number system!

[@kirkanos771]

@@user-qo3qm7ud1d That's not our point. Choosing another base may delay the number of iterations before it crashes.

[@VigEuth]

If you use a different base (non-base 10) will the pattern also break once you get to that base?

[@isavenewspapers8890]

No. The pattern has nothing to do with the digits of the numbers.

[@SgtSupaman]

How does changing the base matter here? The prime numbers are the same no matter what base you use. For instance, just because 15 in base 10 is written as 13 in base 12, it doesn't magically become a prime number. It's still composite.

[@leodarkk]

Well I suppose that one "reason" why you are getting primes at the begining is that this method will never produce an even number, that is guaranteed. It's even weaker than the Paterson method where 2,3 and 5 are excluded as divisors, but it is there :).

[@ThePowerRanger]

That Hollywood celebs analogy was nice.

[@stapler942]

The mediant of two fractions, huh? Is there a submediant? What about a dominant and subdominant? What's the leading tone of two fractions? What's the supertonic?

[@keyaanmatin4804]

How deep does this rabbit hole go?

[@Jkauppa]

analyze the wilson's theorem like the pascal's triangle for each n

[@Jkauppa]

sorry that your brain does not produce clear answers but only mush

[@Jkauppa]

what do you classify A/B/C as a rule, dont you have all as equal gift

[@ChrisSeltzer]

This is why you asked for A list and B list actors on Twitter haha

[@OwlRTA]

lmao, Tim Hein being a very high odd number

[@rainerausdemspring894]

Guy's articles contain some really striking (counter-)examples. I am afraid you need to have access to JSTOR in order to read them.

[@BaccarWozat]

Does the tenth one add up to 33 though? If you count the fact that the number 10 has two digits, you're actually adding 8 instead of 4, making it 37, which is still prime. But there's probably another snag not much further along.

[@GourangaPL]

i came up to a problem with similar thing, start with sequence of 111, each next row is the previous sequence as binary number number XOR itself shifted 1 and 2 bits, so 111 XOR 1110 XOR 11100 so 2nd row is 10101, next is 1101011 and so on, find a way to count how many 1 bits are in the nth sequence, i know for n = 2%k the answe is 3, for n=2k it's equal to the answer for n/2, need a formula for the general case

[@kurtu5]

But what are small numbers? Are the numbers below 2^2^10 small? The largest prim we found is less than that. Are there generating functions like this that work up to something like 2^2^10? And then fail?

[@effuah]

There is mill's constant (numberphile did a video some time ago). It generates infinitely many primes, but the problem is that we can't know this constant to a high enough accuracy without also knowing really large primes. If you want an example for a conjecture that works for small numbers (where the small numbers are really large), look at Merten's conjecture. It has some connection to primes.

[@michiel412]

Just for the record, there's been primes found that are much larger than 2^2^10. 2^2^10 (or 2^1024) has 309 digits, the current largest prime found is 2^82589933 - 1 which has 24862048 digits.

[@Anonymous-df8it]

@@michiel412 I think that 2^2^10 might be the phone number calculation limit as it can only go to x*10^308.

[@ygalel]

1:53 MIND BLOWN

[@LegendaryFartMaster]

2:10 As a certain suspender wearing Frenchman would say: "Today, we're looking at frraacctions"

[@pleappleappleap]

I wonder how the performance of this stacks up against the Sieve of Eratosthenes?

[@guillaumelagueyte1019]

I'm only halfway through the video, but does this mean that the gaps between consecutive primes depend somehow on whether the ranks of the primes are prime numbers themselves?

[@guillaumelagueyte1019]

Oh no the pattern breaks down :(

[@beforeikillyou7430]

His voice🔥

[@IamBATMAN13]

Third part where?

[@Pumbear]

@4:04 Funny addition sign p prrrrrimeee

[@davidlees2963]

Although if you count the both digits of 10, you add 8 which takes you to 37. Then for 11, you have to add 22, that goes to 59. Then 12, you have to add 8, that goes to 67. But it breaks with 13, as you have to add 26 and that goes to 93 (not a prime).

[@Anonymous-df8it]

I wonder which number base produces the most primes?

[@SgtSupaman]

@@Anonymous-df8it , changing the base doesn't matter here. A prime is a prime, regardless of what base you are using (so the pattern is exactly the same, just with different looking digits). For instance, 17 in base-10 is written as 15 in base-12, but it is still a prime number either way, because 15 base-12 has no factors besides 1 and itself.

[@Anonymous-df8it]

@@SgtSupaman If you written ten in base 16, then you'd only need to write 4 digits rather than eight. So that could change it from a prime number of digits to a composite number (i.e. 37 -> 33)

[@SgtSupaman]

@@Anonymous-df8it , except that's not how the pattern works. You consider every number to itself, not its individual digits. When it got to 10, he added four, not eight. By your logic, Grant should have said it continued finding primes at 10 in base-10, but then the pattern falls apart entirely on the next line because you are only looking at single digits, so you don't get 11 everywhere you're supposed to get it (meaning you won't be getting anything related to ϕ(n) once n > base).

[@Anonymous-df8it]

@@SgtSupaman If you modify the pattern so that that's how it works, what number base is the best?

[@GanerRL]

part 3 is just never occuring i guess?

[@Mystery_Biscuits]

It did come out eventually