Prime Pyramid (with 3Blue1Brown) - Numberphile

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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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Videos by Brady Haran
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Special thanks to our friend Jeff for the accommodation and filming space.

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5 ноя 2022

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Комментарии : 437

@numberphile Год назад

Part 1 of this three-part interview is at: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-jhObLT1Lrfo.html Part 3 of this three-part interview: STILL BEING EDITED

@FebruaryHas30Days Год назад

First reply

@Anonymous-df8it Год назад

Second reply

@volodyadykun6490 Год назад

Still

@Anonymous-df8it Год назад

@@volodyadykun6490 Fourth reply

@Jono4174 Год назад

Skewes’s number-th reply

@tommihommi1 Год назад

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

@zmaj12321 Год назад

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

@tyle.s9084 Год назад

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

@otonanoC Год назад

Everything in math was invented by Euler or Riemann.

@louisrobitaille5810 Год назад

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

@tommihommi1 Год назад

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

@volodyadykun6490 Год назад

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

@1224chrisng Год назад

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

@birdbeakbeardneck3617 Год назад

Shheeeeeshh

@lonestarr1490 Год назад

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

@dyld921 Год назад

Grant Sanderson is, like, the sexiest mathematician available.

@kadefringe Год назад

I phap on prime numbers indeed

@mxlexrd Год назад

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

@krokorok_ Год назад

@viliml2763 Год назад

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

@mxlexrd Год назад

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

@ophello Год назад

The first video wasn’t unlisted.

@themathhatter5290 Год назад

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

@Rubrickety Год назад

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 Год назад

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

@berryzhang7263 Год назад

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

@leftaroundabout Год назад

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 Год назад

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 Год назад

@@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

@wehpudicabok6598 Год назад

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

@ps.2 3 месяца назад

Oh, what a lot of fifths there are!

@davidgillies620 Год назад

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 Год назад

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

@redtaileddolphin1875 Год назад

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 Год назад

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 Год назад

@@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

@conanichigawa Год назад

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

@ShenghuiYang Год назад

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

@Vaaaaadim Год назад

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

@viliml2763 Год назад

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

@Vaaaaadim Год назад

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

@razlotan7504 Год назад

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

@deadlyshizzno Год назад

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

@highlewelt9471 Год назад

Grant is always such a delight

@MichaelJamesActually Год назад

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.

@ZacThompson Год назад

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

@JamalanJuda Год назад

My two favorite channels coming together.

@hlvaneeden Год назад

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 Год назад

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

@gaborszucs2788 Год назад

@@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 Год назад

@@gaborszucs2788 Can you provide a visual?

@deadlyshizzno Год назад

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 Год назад

I'm still checking!

@deadlyshizzno Год назад

Why is it still being edited 😭

@deadlyshizzno Год назад

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

@deadlyshizzno Год назад

@Uranyus36 Год назад

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 Год назад

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

@Michael-cg7yz Год назад

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 Год назад

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 Год назад

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

@Michael-cg7yz Год назад

@@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

@anoopramakrishna Год назад

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

@neil5280 Год назад

I check back every day for Part 3.

@neil5280 Год назад

Monday was pretty chill.

@neil5280 Год назад

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

@neil5280 Год назад

Happy New Year! 🎉

@fuuryuuSKK Год назад

DEEPER INTO THE VAULT WE GO

@OwlRTA Год назад

ENHANCE

@ekxo1126 Год назад

@@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 Год назад

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

@ifroad33 Год назад

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

@happyelephant5384 Год назад

This collab is legendary

@AllHailZeppelin Год назад

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… 😕

@razer1024 Год назад

Best video in a long while 🎉❤

@jamesepace Год назад

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 Год назад

😃I bet you have already subscribed.

@jamesepace Год назад

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

@ericpeterson6520 Год назад

Is part 3 still in the works?

@joelkronqvist6089 Год назад

Seems like it was published today

@dhoyt902 Год назад

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.

@dkranda Год назад

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

@toycobra12 Год назад

I thought it was the guy from the KFC logo 😂

@jacksonstarky8288 Год назад

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 Год назад

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

@SuperYoonHo Год назад

Awesome video!

@TheFakeMackie Год назад

3b1b is a phenom channel. Great collab.

@johnkonrath1115 Год назад

Loving the trilogy!

@backwashjoe7864 Год назад

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

@Par_and_syv_lovers56 Год назад

awesome collab

@danieluran9555 Год назад

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

@Sajatzsiraf Год назад

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

@kruksog Год назад

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 🙃

@SpySappingMyKeyboard Год назад

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

@deadlyshizzno Год назад

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

@fidgettyspinner3028 Год назад

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

@lucas.cardoso Год назад

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

@arandomdiamond2 Год назад

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 Год назад

Not very efficient for calculating big primes though

@TheEternalVortex42 Год назад

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

@arandomdiamond2 Год назад

@@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.

@xanderalaniz2298 Год назад

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 Год назад

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

@EPMTUNES Год назад

Grants always been a great math communicator!

@a0z9 Год назад

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

@AidanRatnage Год назад

Suddenly, it's not unlisted anymore!

@zerosir1852 4 месяца назад

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

@deadlyshizzno Год назад

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

@JamesJoyceJazz Год назад

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

@FirstLast-gw5mg Год назад

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

@TheCapcarap Год назад

This is the ultimate video

@bumbleandsimba Год назад

NUMBERPHILE I LOVE YOU'RE VIDEOS 💗

@leobarlach Год назад

That's the funny addition video! Classic!

@toferg.8264 Год назад

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.

@abuzzedwhaler7949 Год назад

Papa Grant here to give us some key geometric intuitions

@bad_manbot Год назад

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 Год назад

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 Год назад

@@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 Год назад

@@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.)

@senthilkumaran5255 Год назад

neat sleight of hand at 3:47 :)

@anonymoususer2756 Год назад

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

@kingdomadventures Год назад

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

@addymant Год назад

Will you upload the third video unlisted?

@cloak_poison2124 Год назад

THE CROSSOVER I DREAMT OF

@keyaanmatin4804 Год назад

How deep does this rabbit hole go?

@CorrectHorseBatteryStaple472 Год назад

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

@francescos7361 Год назад

Thanks .

@hyftar Год назад

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 Год назад

it's not a base 10 specific thing

@smizmar8 Год назад

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

@TaranovskiAlex Год назад

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?

@thatoneginger Год назад

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

@kp2k Год назад

super cool

@GanerRL Год назад

part 3 is just never occuring i guess?

@Mystery_Biscuits 11 месяцев назад

It did come out eventually

@bstlang Год назад

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

@ChrisSeltzer Год назад

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

@stapler942 Год назад

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?

@Chunes3 Год назад

Grant called the mediant "not a wholly useless operation" which implies it is partially useless.

@ygalel Год назад

1:53 MIND BLOWN

@shrayanpramanik8985 7 месяцев назад

Now if I say to some kid who watches numberphile,that Jennifer Lawrence was in a numberphile video, would they believe it😂?

@axelnilsson6478 Год назад

Poor Tim!

@pleappleappleap Год назад

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

@countrychurchmonuments7906 Год назад

Never mind all that. I want to know why he has a combination lock on the door in the background.

@rainerausdemspring894 Год назад

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

@OwlRTA Год назад

lmao, Tim Hein being a very high odd number

@LegendaryFartMaster Год назад

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

@GourangaPL 5 месяцев назад

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

@leodarkk Год назад

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 :).

@beforeikillyou7430 Год назад

His voice🔥

@btf_flotsam478 Год назад

I wonder if the numbers from this sequence satisfies Gilbreath's conjecture.

@15october91 Год назад

3Blue1Brown is the GOAT.

@Jkauppa Год назад

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

@Jkauppa Год назад

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

@Jkauppa Год назад

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

@BaccarWozat Год назад

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.

@SebWilkes Год назад

Worth pointing out the attention thing that RU-vid displays skyrockets when the actors are shown on screen lol

@Verlisify Год назад

Fun video

@chessandmathguy Год назад

I'd like to see a seemingly true conjecture that was thought to be true for a long time until someone came along and definitively proved it false. That would be something.