63 and -7/4 are special - Numberphile

Dynamical sequences, prime divisors and special exceptions.
More links & stuff in full description below ↓↓↓
Featuring Dr Holly Krieger from MIT.
More primes: bit.ly/primevids
More videos with Holly Krieger: bit.ly/HollyKrieger
Since this was filmed, Holly has become a mathematics Lecturer at the University of Cambridge and the Corfield Fellow at Murray Edwards College.
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Videos by Brady Haran
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[@ClickBeetleTV]

It's always comforting to see that talented mathematicians still have a momentary dip in confidence when it's time for arithmetic in public

[@dannygjk]

It's like after people get their driver's license most of them forget how to parallel park.

[@SoumilSahu]

best handwriting on numberphile

[@manoelnt0]

Really pretty.

[@jans1982]

Not just the handwiting.

[@zwitter689]

best hands!

[@What-thaW]

hands down

[@the_original_Bilb_Ono]

Bottoms up

[@careernerd2135]

"So it's all Mandelbrot?" "Always has been."

[@McCaffreyPickleball]

Is that "Never Back Down"?

[@FenceThis]

That's Numberwang

[@googlestore4830]

Every time Ms Holly laughs, a new kitten is born.

[@ECSSANJAY-lr2hu]

Ser

[@leadnitrate2194]

@@ECSSANJAY-lr2hu was that a joke referencing "white knight"? If so, well done.

[@reedplaysgames]

Statistically speaking, probably yeah.

[@PayDaVig1]

Haha. I'm a believer.

[@megamillionfreak]

My thoughts exactly.

[@dhampirmike]

Seeing someone just freehand the general shape of a fractal like that is quite impressive.

[@sasdagreat8052]

Had a random bout of obsession with the Mandelbrot once, and learnt the dimensions of the fractal as a consequence. It's a fun shape to doodle when bored, and instantly recognisable by any passing math enthusiast.

[@kingditto5149]

How to get rich: live in the UK, sell brown paper and sharpies

[@angelicaabengosa3077]

Anglcaangelics

[@christianerbgarten5057]

Mbta problems

[@EHMM]

Numberphile: *i'll take ur enitre stock boi*

[@swagatata]

like David Brent?

[@munjee2]

@@EHMM the more British version would be "we'll take the lot"

[@ILikeWafflz]

4:28 Well that escalated quickly.

[@willisverynice]

+ILikeWafflz Exponentially even.

[@joelsookdeo7498]

Yep

[@Ken.-]

I saw that. Brick killed a guy.

[@Axileoni]

hahahahahhahahahahha lol

[@strangelyjamesly4078]

Bravo

[@numberphile]

Brown paper from this video: cgi.ebay.co.uk/ws/eBayISAPI.dll?ViewItem&item=380872542159

[@SinanAkkoyun]

Please sell more brown papers ^^

[@SinanAkkoyun]

OrphanPaper xD

[@mbrsart]

Should have gone for £63. ;)

[@mayankacharya2712]

Numberphile::: you missed to present f(x)=1/2{6x-(-1)^x+3}. This gives more prime numbers.

[@Sigmav0]

mayank acharya Really ? Where did you see that ?

[@jaimebenito620]

I've got the feeling that, all of a sudden, a lot of people are going to become very interested in maths.

[@franciscopais6200]

Glad someone else noticed :D

[@SawSkooh]

I know, right? An American for a change.

[@y788lhjk1]

what university is this I need to apply, wanna study math.

[@DanHarkless_Halloween_YTPs_etc]

Yeah, how did this channel go about cornering the market on brainy, beautiful, freckly redheads and strawberry blondes?

[@Huels]

Math has always been sexy. It's the language of the universe. When I was getting my Masters at Columbia at the business department, the ladies were getting it done.

[@stottification]

7/4 is a pretty rad time signature! I suppose -7/4 would mean the song has to be played in reverse.

[@alexanderkonczal3908]

+Scott Lee absolutely rad. let's see if people know the (arguably) most famous song in 7/4...

[@CodyMudrack]

+Alexander Konczal Probably something by Dream Theater, I'm sure lol. Oh wait!! they play in 9.2/5. thats right...

[@alexanderkonczal3908]

what abouuuut... MONEY by pink floyd! ha! such a well known song, but people don't think about it.

[@CodyMudrack]

ahh. Haven't heard much my Pink Floyd so that's probably why I didn't recognize it

[@alexanderkonczal3908]

I have to correct myself - I originally wrote time, I meant money. I guess in my mind, the saying "time is money" is true.

[@ibbz6932]

Is there anything Amy Adams doesn’t know how to do?

[@babydriver8134]

Raise babies?

[@joshgibson267]

B.D what??

[@luciano53688]

Winning an Oscar.

[@jacobschiller4486]

@@luciano53688 oooof

[@baktashzameer4193]

I think that's Isla Fisher

[@williamn1055]

Am I the only person who heard a chicken at 6:46?

[@bentoth9555]

I've watched this several times and never heard that before. But it is definitely there.

[@adampowell3165]

its a squeak in the desk that the paper is on

[@bentoth9555]

That does seem more logical than a random chicken.

[@kingkoy7397]

+Civil Engineering Philosophy taga um ka Noah?

[@mud2479]

lol

[@Tfin]

It seems like every number is special. What number is _not_ special? Because _that's_ the special one.

[@madsskipper9408]

Lincoln Lopes KZbin is on the conspiracy!

[@saumitjin5526]

If all numbers are Special, what makes an individual number special? All of them are special so it must be "Common" really (to be special).

[@MalcolmCooks]

this is known as the smallest boring number paradox

[@servandopineda1878]

For sure all positive integers are special! Assume not. Then there exist a set of non-special numbers that has a minimal element. But hey that smallest element is special because it's the smallest! Contradiction. Thus all positive integers are special 🙃

[@michaelantoun9353]

A famously known proof by contradiction that every number is, indeed, special!

[@Valto4life]

How to heck did she draw that graph without grid paper? She must be a wizard.

[@DeViLTh0rn]

Dustin Boyd 🤣🤣🤣

[@geewilikers9780]

"It might be a harder question, depending on how specific you want to get." A truer statement never told. :)

[@elietheprof5678]

65535 is interesting because all the prime factors are one more than a power of two (3, 5, 17, 257)

[@antosha4812]

I know this comment is ancient but I want to point out that all numbers of the form 2^2^n - 1 have this property (except that these divisors may not be prime). They will be divisible only by all numbers 2^2^m + 1 for m < n. It's not super hard to prove either :)

[@antosha4812]

However, it's an open question whether 2^2^n + 1 numbers are prime for infinitely many n, or even whether they're composite for infinitely many n. Heuristics strongly suggest 2^2^4 + 1 = 65537 is the last one that's prime.

[@johandenbakker2513]

It's moments like these that I'm proud to be studying math. This sounds a lot like the useless "tinkering" I did in highschool to get through the boring hours. Nice to know that there are very intelligent adult people who, instead of going "why would that ever be usefull", say "hey, this is peculiar. Let's see what happens when I do this".

[@Teathymeterror]

And for those wondering but too lazy to do the figuring, x^2 - 2 ends up being -2, 2, 2, 2, 2, 2, 2, etc.

[@lpcamargo]

Watching Numberphile is always awesome but sometimes also surprisingly relaxing too!

[@TheSentientCloud]

Oh, and fun fact. Mandelbrot literally means "almond-bread" in German.

[@jeremiahseitz9842]

If this is indeed true, I appreciate this trivia. :)

[@TheMrvidfreak]

We meet again, Mandelbrot set, and you never stop surprising me.

[@trevorhuff7213]

Man, if someone would've shown me this channel when I was in high school I would've liked math so much more. Maybe I'd be somewhere better than community college :( Oh well, at least I've learned to love it now.

[@markmiller6402]

I have no idea what I just watched. But that lady’s enthusiasm and intelligence is amazing

[@clairespivak2953]

I love how f(x) = x^2 -2 gives you a square root sign when you plot the results

[@ankitmeena6842]

underrated

[@LunizIsGlacey]

True, didn’t think of that!

[@elliottmcollins]

This was fascinating. Any more footage explaining what exactly the Mandelbrot set is would be great.

[@yvesdelombaerde5909]

There is a video (not Numberphile) linking the M-set to Feigenbauw constant and the behaviour of the Xn+1= r.Xn.(1-Xn) eq.

[@singami465]

I know Numberphile does a lot of simplifications for beginners, but still, the misuse of functions here is really bugging me out, especially that it's also going to confuse a lot of newbies too. Getting powers of two is simply "f(x) = 2^x" and "missing" that by a one is, again, simply "f(x) = 2^x - 1". You're talking about sequences (so variables like a, n, q and r), yet you use a function, which is really weird.

[@jialixx]

Numberphile has become my favorite channel now, I watch it every day and can't stop. Thank you all for those awesome videos.

[@Saber-tooth]

At the beginning of this video, it was stated how the series generated by f(x) = 2x + 1 was always equal to 2^n - 1. If you perform the function in binary, rather than decimal, it becomes clear why. The function f(x) = 10x + 1 [or f(x) = 10x + 9] would have a visually similar effect in decimal.

[@samueldeandrade8535]

I guess f(x)=10x+1 is the function that truly would have the same visual effect. Namely, the terms would be 1, 11, 111, 1111, ... for each corresponding bases.

[@ZoSoxSwAnSoNgx]

Has James Grime done something different with his hair? He looks different in this video...

[@olliefoxx7165]

It's the shirt, I think.

[@Commandelicious]

Amazing handwriting.

[@agokhandemir]

Best handwriting in the series so far...

[@supermegajaime]

Seems like women have prettier handwriting than men.

[@michaelbauers8800]

I have noticed same. But I remember one dude with handwriting I thought was of female origin. I did a double take, mentally, heh

[@hayricandurmus4467]

Moruk turksen turkce konus

[@bluesky6905]

Hayrican Durmuş Türklere mi konuşuyo

[@hayricandurmus4467]

Moruk o zaman yazmayacak bir sey

[@Alex_Off-Beat]

It's actually completely mind-blowing that -7/4 is an exception is to this rule considering that even if there were an infinite number of exceptions within the Mandelbrot set, the fact that even one of them is a rational number is unbelievable, since the Mandelbrot set only contains a countably infinite number of rational numbers whereas there are an uncountably infinite number of irrationals within that set. Meaning even with a countably infinite number of exceptions there's effectively a 0% chance that any of them would be rational...

[@jonathanschossig1276]

Whole numbers are also rational.

[@Alex_Off-Beat]

Yes they are, what's your point?

[@hussaingamer4163]

Whole numbers are pretty

[@adron2532]

Wouldn't any set containing an interval of the real line contain countable rationals and uncountable irrationals?

[@OskarElek]

I understand your amazement, but keep in mind that irrational numbers, as interesting as they are, are completely irrelevant in this discussion - there's nothing like prime factorization of an irrational number, or even a denominator to speak about...

[@LittleCD]

Dr. Krieger seems like a good fit in Numberphile! Hopefully she does more videos in the future :)

[@mdsharfuddinmd5710]

Thank you sir

[@jona5820]

I come home, tired after a day with to much math at uni. To relax and to get my mind of work, I watch Numberphile.

[@PoliticalJohn]

I think people don't realize how much work goes into your videos. Thanks Bradypus!

[@warningsigns4526]

brains and beauty

[@Hythloday71]

Brill as usual. But eagerly awaiting a SIXTYSYMBOLS on the recent gravity wave discovery confirming inflation ?

[@Wildpfad]

It'd be cool if you showed the proof that those sequences will always have new prime divisors if its not too complicated.

[@patrickhodson8715]

I just tried _f(x) = x² - 2_ and when I went to calculate the fourth element in the sequence I actually lold. Well played, ma'am.

[@richardprice6881]

So impressive...and people are sharing videos of kittens. This is youtube gold and has me love the internet again. Thank you.

[@brunoramirez2602]

Richard Price hey, kittens are great!

[@MichaelFrancisRay]

She's great. Hope we can see some more numberphile feat. Dr kreiger

[@ZeroKelvin]

That giggle after "if your rational number is very, very close to that special point in a technical way that's hard to formulate." is incredibly cute - largely in part to the intelligence that proceeds it.

[@sidefish8362]

If you zoom into the Mandelbrot set at a point on the real axis which corresponds to -7/4 (i.e -1.75) you come to a needle-thin point at the very innermost tip of the split in the bulb on that mini Mandelbrot. You can keep zooming in to that tip but you can never "arrive", no matter how many times you increase the iterations, it's like you have reached an infinity point. I dare say that there are an infinite number of these points along the Mandelbrot set's "real" axis which appear at the same point at the very tip of the split in the bulb of each minibrot, of which there are an infinite number.

[@lydaloudon3338]

You should definitely do more vids with Dr. Holly Krieger!

[@lxathu]

Every time I start being a bit afraid Numberphile will burn out after all, it suprises me with something really new to me. Thank you to all creating and participating in this exciting series - and its siblings as well, of course!

[@fmontpetit]

Great video! Dr Krieger is fantastic! I hope we see her again!

[@JohnDoe_69]

09:12 I'm in love.

[@dengland5874]

Yep!😍

[@ryleexiii1252]

I love going along with my own math while watching these videos. Makes for a fun time.

[@epicpurple129]

*looks on the chalkboard in the background* wut

[@ZardoDhieldor]

It's always great to see someone new in Numberphile! And this is a fascinating topic as well! :)

[@jeremiahseitz9842]

Your username is Moon Rabbit? Not slagging. Just think it's cool.

[@timurkotulic3948]

Technically, it is "The Hare of the Moon", rabbit is "cuniculus"

[@KutuluMike]

I still really want to know why it matters that 63 was the 6th element of the sequence...

[@ISpaceGhost9I]

+Michael Edenfield All multiples of 3... ILLUMINATI !

[@ranged12345]

+Michael Edenfield Zsigmondy's theorem

[@Eggemeyers]

Me too! I do know that 63 is special because it is the smallest whole number that can be divided by all whole numbers between 1 and 9 without producing a remainder. No clue what the significance of it being the 6th element in the sequence is about though :(

[@ISpaceGhost9I]

Kyle Eggemeyer I don't think that I understand your comment... There's something wrong here "63 is special because it is the smallest whole number that can be divided by all whole numbers between 1 and 9 without producing a remainder" right ? Or am I just too tired to understand what you meant...

[@htmlguy88]

+Kyle eggmeyer I second the previous thought about your statement 63 =9*7 it doesn't divide by any numbers other than 1,3,7,9,21, and 63 ?

[@johannesh7610]

With the fractions, you could also multiply by the denominator and then ignore it as a common factor (and not a new prime). By that you also encompass ax²+c, where a and c can be rational

[@TamNguyenphoto]

These sequences are important because they have the ability to generate the next biggest known prime, which is fundamental for your HTTPS/SSL, which uses RSA encryption, which relies on having 2 big primes. The bigger the better.

[@davidm9442]

Interesting stuff i see

[@007MrYang]

I'd like to see more about this sequence.

[@olmen375]

Intelligence is beautiful. I hope we get to see more videos with Holly.

[@mazxbv]

what are your thoughts on Grigori Perelman?

[@jamespfp]

Excellent and informative, TY Holly and Numberphile!

[@PrimusProductions]

(x-63)/2=3

[@ahuddleofpenguins4842]

nice

[@RhapsodyAfternoon]

I'm disheartened that everyone seems to be commenting exclusively on how attractive she is. Can a good-looking woman discuss mathematics and spur a discussion on the actual topic like all of the other Numberphile hosts?

[@definelowl9775]

Is there anything wrong with appreciation of look?

[@ironDsteele]

No.

[@GeorgiosB]

Special kind of trolls out there. And the marriage proposals... oi!

[@donho1776]

Sexual attraction is a reality of life. We should not be ashamed of it or feel we must apologize for, or not admit to feeling it.

[@peterwestberg9894]

get over it.

[@meteormedia7021]

Wow, Dr Holly Krieger is a stunner:)

[@DiveDiva2007]

I did maths at London Uni (Kings College ) in late 1960s and still find this stuff fascinating- thanks

[@Fogmeister]

I could listen to her all day. Love the maths and the explanations!

[@paulflute]

more mondelbrot math vids please.. and Julia..

[@JasonRudolphKruger]

I am kind of off topic but 4 divided by 7 = 0.57142857142857142857142857142857 times 63 = 36 (63 mirrored) I thought this was the point of the show before I watched. Oh yeah this also works for 84, 42, 21, and probably others (like 4284 is 2448 and 5628 is 3216) notice the second digits or second group of digits are 1/2 of the first one in all cases. Well thought I'd share thank you!

[@EmoJakeInJapan]

her circles are amazing

[@naota3k]

You should do a video about the different fractal sets. I would like to understand what exactly they are.

[@razzledazzle1233]

I wish she was my math teacher.

[@rothgang]

I don't. I wouldn't learn anything.

[@blinkcatmeowmeow8484]

Conner Trieskey why?

[@helvecioresendechaves]

Conner Trieskey 🧐😆😆😆😆 I got that

[@blinkcatmeowmeow8484]

Helvecio Sniper wait lol Ive just realized 😂😂

[@Kris.G]

@@rothgang me too and it's quite pathetic really...

[@rish1459]

I never thought of trying to work with this type of iteration! Thanks for this...

[@sbarr10]

Was there a switch here? In 2x + 1, Dr. Krieger worked through the positive integers sequentially: 1, 2, 3, 4, etc. But for other sequences it looks like she plugged in as x the previous output.

[@nickplatypus1730]

what happens if you use Pi or Phi?

[@Biggnuncio]

While many of these mathematical patterns are sort of interesting I always wonder how much time and effort has been put into them and what value has come out of that work. I wish one of your questions on these was always "How can this be applied to the real world?" or "Now that we know that what else do we know?"

[@thegreatreverendx]

The math is interesting. But what I also love about this video is her voice. It’s confident, lyrical, clear, tempered, and articulate, like the way Americans spoke back in the 60s or 70s.

[@joey86bu1]

A cute red head and numbers, this is my fav't video of all time!

[@SPEED20011978]

I don't know if Numberphile has a video on this topic but I think you should make a video on why any number to the power of 0 equals 1

[@breadnoodle]

laaaaaaate replay Well, nearly all (poor 0 :c) x^(0) = x^(1-1) = (x^1)/(x^1) = x/x = 1 (plus apparently there are 20 or 11 comments in here *but* I can't see any for some reason so idk if it has been already written)

[@5up3rp3rs0n]

f(x)=x^2-2 0^2-2=-2 (-2)^2-2=2 (2)^2-2=2 And loops forever

[@Riseky]

Thanks

[@AaronHollander314]

Are you sure?

[@dingovory]

Aaron Hollander it turns to 3 right after infinity

[@maximumdosage]

Thank you

[@starchild2299]

i did that in my head in 5 seconds, am i smart enough yet?

[@golux-57]

Fantastic! Such interesting analysis!

[@snorrebraathenkjeldby7316]

Amazing voice for narration

[@DavenH]

Nice speaking voice, sounds like a professional broadcaster. Then again lecturing is good practice.

[@andreanordick8253]

So interesting! I'd love to see more about fractals and the Mandelbrot set!

[@jeremiahseitz9842]

I first read that as "freckles". 🤣

[@fieldrequired283]

She has a really lovely voice. A joy to listen to while I was driving home from work today.

[@yaboi3339]

what a calming voice

[@ButzPunk]

I tried -2 and it made me laugh.

[@karlkastor]

me too too too too too... :)

[@jwhalstrom75]

Why exactly? I tried out the sequence but perhaps I calculated it wrong.

[@karlkastor]

It's 0, -2,2,2,2,2,2,2,2...

[@bigglessy]

...I think I saw a 2 D:

[@NiramBG]

lol I tried it too, it's definitely funny.

[@shenkeey]

7:19 I keep hearing it as Bill Cosby ''You can ask this as a fraction, see? Instead of a whole number, see?''

[@simicpetar]

Petar Simic 0:46 "This sequence is special..."

[@shenkeey]

***** too soon... (but secretly I laughed)

[@AEARArg]

That being said, wonderfully explained topic!

[@rainbowspite2221]

Mind blown. I love all the feelings of awe Mandelbrot gives me.

[@FelkniaMusic]

This is a proof that Ygritte knows more things than Jon Snow !

[@tabularasa0606]

Everybody knows that John Snow knows nothing.

[@FelkniaMusic]

tabularasa0606 At least he knows that one thing with the thong

[@mikestevens8012]

" something messy that I don't want to calculate" wow ! I didn't know that was a thing ,option ...

[@jorgepeterbarton]

Ginger based racism

[@davidsweeney111]

do these dynamical properties of numbers extend to effects in physical systems? Great video, really like Holly. Please make more ....

[@StreuB1]

....we need more women mathematicians. It was very refreshing listening to her explain this. And shes from the town right next to mine; Illinois girls FTW!

[@ryan0348]

We need to see more of her.

[@razorborne]

would've been nice to have some sort of explanation as to why numbers in specific points on the Mandelbrot set are special. I understand the math is probably fairly difficult, but at least an overview.

[@martincohen8991]

I agree. This is the really surprising part to me. Would love a link.

[@19seb85]

well now I want an extra video about why 6 is special :) also she has a great voice

[@NoPrefect]

what a triple threat! brilliant, beautiful, and charismatic!

[@n124ac9]

6:01: The reason why is because that sequence leads to: -2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,…

[@raglanheuser1162]

i cant tell sometimes if primes are some weird thing humans have this fascination with or some massive universal truth we have only scratched the surface of

[@godsamongmen8003]

At least in some areas, primes are useful tools. I'm sure they also represent some universal truth that nobody knows yet, but for now they can make our encryption keys.

[@brendanh8193]

They are a universal truth about a particular complementary concept. They define what is NOT, rather than what IS. (That is, they are defined by not having a factor other than trivial factors.) I wonder if there are other concepts that are defined by "nots"?

[@teemuaho4807]

@@brendanh8193 I mean you could define a prime as having only 2 factors

[@stuartdparnell]

Mandelbrot sets are connected to logarithmic spirals so yes you're not far off with that statement about special universal truth.

[@jonasdaverio9369]

@raglanheuser The fact that prime numbers give a unique decomposition of any number gives a clue about why they are fundamental. But of course, there are other far more advanced or fundamental things in number theory that involve prime numbers that I know absolutely nothing about. But, yeah, they are not just weird random artifacts.

[@Zreen001]

I didn't know Ginny Weasley was a professor at MIT...

[@colorcookie6088]

😂😂😂 lol

[@reinaldofanuel5892]

Your explanation is Amazing

[@dasdos002]

Very nice video. Great math on the blackboard too!

[@gonzotown9438]

-7/4 = -1/1-1/2-1/4 = -1/4-2/4-4/4 That feels important somehow.

[@ellock1998]

Shouldn't 0^2 = 1? I remember that being a property or something... I think. So shouldn't 0^2 + 1 = 2? Or did I dream some nonsense up...

[@tonelemoan]

Other way round: 2 ^ 0 = 1.

[@leviathoncannon]

something i took from this is that, if a thing has some property that we define, it is interesting. And they complement the uninteresting cases. I also learned that sometimes, we will find inconsistencies in these mathematical games we play, quite like glitches. Those glitches may point to deeper truths, is maybe one way you can put this concept

[@eixpata502]

Thanks for the video/explanation.