Primes are like Weeds (PNT) - Numberphile

The Prime Number Theorem shows that primes are like weeds, popping up everywhere! Dr James Grime explains --- Little bit extra cut from this video: • Prime Number Theorem (...
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[@pamanes7]

you should do a video where he explains his Phd thesis to us mortals

[@CraftQueenJr]

Yes.

[@MK-13337]

Original comment is 4 years old which is fitting since I asked him about his PhD when he visited our school 4 years ago. It was about combinatorics and linking combinatorial math to matrices and linear algebra. I think you can find his PhD on the internet (maybe I havent searched for it)

[@shoutz5872]

@@MK-13337 Every PHD thesis is aviable for free

[@MK-13337]

@@shoutz5872 All PhD thesis are in principle available but not all of them are in online archives. But if you go to the department where the PhD was from they have it on hand. As I said I didn't check

[@Causticghoul]

The internet needs more James Grimes.

[@sliceofgarlicbread6868]

He has his own youth channel, it's called singingbanana.

[@sliceofgarlicbread6868]

"youtube

[@OceanicEdits]

hahahah youth channel i was gonna say xd

[@femioyekan8184]

The world needs more like James Grimes.

[@oz_jones]

James Primes more like, amirite?

[@ChrisBandyJazz]

How about you just change your name to James Prime?

[@shaurikdeshpande1889]

He gets high on maths. He even thinks primes are weed.

[@SoberCake]

*Optimus

[@CaseyShontz]

Chris Bandy he’s going to marry a sexy prime some day

[@00bean00]

Prime, James Prime. Agent 00mod7

[@wurttmapper2200]

00bean00 Your comment is underrated

[@tubrutolity]

Read the title and was like "Smoke primes everyday"

[@TheReligiousAtheists]

Hey I was going to comment that joke

[@TheLeonroi]

3 years too late

[@andrewxc1335]

Anyone else watch Lurd of teh Reings? That joke is in there a lot.

[@Krong]

never gonna prime you up

[@traktortarik8224]

419 and 421 are twin primes

[@pfl95]

I actually find this site more interesting than 12 years of elementary to highschool education.... the comments are great too. People discussing about this and that. Makes young audiences interested in maths. I hope teachers use this channel

[@leepoling4897]

Being a 9th grader who is quite interested in these videos I think it would be very beneficial if other kids my age watched these type of things. This channel made me actually enjoy math.

[@yea9008]

/r/iamverysmart

[@myriadsfeynman9096]

I just can't agree more with this comment. It should really make us think about the education system.

[@snelo67]

The constant e can be remembered by using the following: Andrew Jackson was president of the USA in 1828; and the angles of an isosceles right angled triangle are 45,90,45. So remember 2.7; Andrew Jackson; Andrew Jackson; isosceles right angled triangle That is: 2.7 1828 1828 459045 which is e to 16 decimal places

[@hansb1337]

thats realy cool!

[@krischurch5677]

+Andrew Snelson Hi Andrew... Is there a way to determine 1 to more decimal places ? - is there a way to choose how many decimal places you want to, to determine 1 ? ( i know 16 is a lot but im looking to take it further - thanks

[@andrewsnelson6794]

+Kris Church Not that I know - e can be calculated - but it will be easier to look it up - I just learnt the quick memory tool to remember it to 16 places. Which should be accurate enough for most real world applications

[@krischurch5677]

Ah ok.... i need to go as far as 25 places for a study you see. and the mechanism of calculating would be useful for any adaptations / conversions. will try looking it up. Thanks for the reply

[@TheVMDC]

Actually you can calculate it further if you study history, as well phone numbers.

[@StuziCamis]

2 x 2 x 3 x 5 x 7.

[@grantmacdonald3904]

They're like weeds

[@stevejobs5488]

Smoke 🌿 everyday.

[@randomdude9135]

420

[@hewhomustnotbenamed5912]

3×23

[@sortagoodish8491]

@@hewhomustnotbenamed5912 nice

[@Magickmaster3]

It is funny that when i started watching numberphile, i didnt understand anything and now i understand EVERYTHING they say!

[@WalrusRiderCycling]

Look at the video on Godel lol

[@IntimateMuffin]

If anyone is interested by this video, I highly recommend the book "Prime Obsession" by John Derbyshire. I read through this book as a senior in high-school, and even though I did not fully comprehend the proofs of the theorems presented, it was a great read and really enhanced my problem solving methodology. The author elaborates on Bernhard Riemann and his Hypothesis, and the Hypothesis' intimacy with the PNT. Every other chapter also includes history of the PNT and it's contributors.

[@PopeLando]

I recently found this useful when discussing cryptography. RSA cryptography (simple) creates an asymmetric cypher by providing a very large unfactorisable number (ie the product of two enormous prime numbers) with which you perform a modular exponentiation. Currently a lot of implementations use 1024-bit prime numbers to build the cypher number. So if you were trying to find prime numbers represented by 1024 bits, how many prime numbers is that? Well, base-2 log of 2^1024 is 1024. e is between 2 and 3 (closer to 3) so the natural log of a number is likely to be approximately 2/3 of the base-2 log. But in any case, base-2 log of 2^1024 being 1024, we know that "pi" is going to be no smaller than 1/1000 of 2^1024. Well, if you have a calculator that can handle large exponents (eg MS Calc for Win10 can) you'll find that 2^1024 is about 1.8x10^308. ln(2^1024) is about 710, and so pi(2^1024) is 1.8x10^308 / 710, which is 2.5x10^305. So the PNT tells us that in the realm of 1024-bit numbers, ie 10^308, the number of primes is 10^(308-3) or a still massive 10^305.

[@youssefdirani]

Is it 1.8 x 10^308 x 107 or / 107 ?

[@PopeLando]

@@youssefdirani You read all that? The number of prime numbers less than 1.8x10³⁰⁸ is not going to be *bigger* than 1.8x10³⁰⁸, is it? π(n)=n/log(n), log(1.8x10³⁰⁸)=710. 10³⁰⁸ divided by approx. 1000 = 10³⁰⁵. In other words there's still a gigantic number of prime numbers to choose from.

[@naveenchandrakumar480]

Most important thing I like in you is the amount of enthusiasm you have to know about the properties of these numbers. Great explanation of the PNT.

[@kapitan_black_pearl]

this title is so right, every time I see prime numbers i get so high. there is no multiple to explain this euphoric feeling

[@DanielEyassu]

These videos just blow my mind every time. Thanks Brady and Dr.Grime

[@WalrusRiderCycling]

Wow I actually could follow that . Yay..

[@MadNotAngry]

Only understand about half of all your vids, Numberphile... but enjoy each and everyone.

[@Infinitesap]

I really enjoy all your videos. Im about to get really motivated to envestigate. Thanks - and please don't stop.

[@elibaum6648]

When I first saw the title, I thought it said, "Primes are like Weed"... lol

[@MrSmith2100]

Math hasn't been the same since I had a chalkboard moved into the bedroom. My math has been longer lasting, more energetic, and better over all. That's funny, because it's probably going to ensure I never have sex.

[@ChristAliveForevermore]

Euler's constant is absolutely extraordinary.

[@putinstea]

But can you roll a joint of primes?

[@anshmehta7613]

Iorveth look up ulam spirals 😂

[@TuhinDas]

Why dont u do videos with JAMES GRIME anymore Brady? His videos are great.. So simple explanations

[@rfrydell5430]

He's a very likable guy and he's ok. But he's not really just ok.

[@0LoveSong0]

Dr.Grime kind of looks like an ostrich in the thumbnail.

[@BIBLE-a-s-m-r]

I’m smiling from ear to ear because I’m in the edge of my seat

[@sigma4805]

These prime number video's are fantastic! keep it up!

[@CatnamedMittens]

Smoke primes everyday.

[@grantmacdonald3904]

Lel

[@tombackhouse9121]

421 every day

[@TitleistGuy]

I love mathematicians. As an engineer I always thought I had a handle on math but honestly thats barely scratching the surface and these guys and gals on this channel are the people that really get math.

[@PeterPrevos]

It is the composite numbers that are the weeds.Primes are a thing of beauty.

[@thekerchmar4462]

Nice clarity on the tilde!

[@PhilBagels]

That means there is at least one prime between Graham's number and 2x Graham's number. So all you have to do is search that limited interval, and you'll find the biggest prime so far! So get to work!

[@CraftQueenJr]

No.

[@eoghan.5003]

Ah yes, the limited interval that is a Graham's number large

[@SN-dy4rp]

So you are very good with numbers. My favorite number is 3. I've been taught how to find phi by using prime quadruplets. 1st take your 3rd (you could use any of them) 101, 103, 107, 109 and the 4th 191, 193, 197, 199. Then assign a number in the middle: 105 and 195 (101,103, {105}, 107, 109) and (191, 193, {195}, 197, 199). the assign the { } number a prime sequence number. 101 being the 26th prime and 103 being 27th, 107(28th), 109(29th) ... 191(44th prime) 193(45th), 197(46th) 199(47th). Since 105 and 195 ARE NOT primes we have to assign a sequence number so 105 being 27.5th and 195 being 45.5th. Then take 44.5/27.5=1.618. Magic? My question to you is we are a extremely intelligent race of animals(humans). But yet our technology is merely rediscovery something that was already there. Numbers of mathematical fundamental, constant anywhere, and this cyclical nature of number and science. Is it just random chance? Or was it created? Just like your thoughts.

[@volbla]

That other theorem you're thinking of doesn't say that the largest gap between primes is 70 000. It says that however high you go on the numberline, there will always be a couple of primes that are separated by less than 70 000. Most primes at that level will still be separated by more than that.

[@SomeMathematics]

Your question is important. And yes, in an infinite way, there is a bijection from N (the positive non zero integers) to Pn. This is easy to see by setting a function such that f:N->Pn, with formula f(n)=pn, and it is easy to prove that the function is both injective and surjective. So it is a countable set. Unlike R (real numbers) the set of prime numbers is the same infinite size as N.

[@AmeeliaK]

If he had been my teacher when I was a teen, I would have been so in love.

[@elzoog]

What he says though has to be balenced by the fact that you can have a gap between primes as large as you want. To see this, consider the factorial function n! = 1*2*3*4*5*6*..*n If I want a gap between primes to be, say 100, take 101! Obviously 101! + 2 is going to be divisible by 2, 101! + 3 is going to be divisible by 3 ... 101! + 7 is going to be divisible by 7. So we have all of 101! + 2, ... 101! + 101 all being composite and thus we have a gap between primes of 100.

[@00bean00]

That may be a true bound, at that magnitude, but there are smaller primes separated by the same bound. You can divide n!s by 2,3,5..p to get n# ("n primorial"), and those are your smallest numbers to start from. In other words, it is sufficient but not necessary.

[@matteo-ciaramitaro]

well if n doesn't have to be a prime then you have the whole set of negative numbers and 0 to work with, assuming it is a real number. In which case there are more examples where there is not a prime in between than there are examples containing a prime between. The actual postulate states that n>3 and it is n

[@NotthatRossKemp]

Great video on primes numberphile!

[@thekkl]

TIL tilde's have more of a meaning than simply approximately.

[@ThisUserHasBeenCanceled]

Why would you multiply [the average gap up to N] by [N] to get the N'th prime? Doesn't he mean the average gap up to the N'th prime? The average gap between primes up to 135,221,143,753 * 5.500.000.000 = 140.965.975.573, which is a lot closer. Could someone please explain?

[@couplabeersnobeers]

+T Geijtenbeek This confused me too. It doesn't make sense to multiply the average gap up to 5.5 billion BY 5.5 billion. That means you are saying the first 5.5 billion primes are separated by an average of ln(5.5 billion). But according to the PNT the first 5.5 billion primes are separated by an average of ln(135 billion). If I had to guess it's because you will have two unknowns in the equation if you don't know the prime numbers. Therefore you can substitue the prime number itself with the number of primes (by using n for both). As you approach very large numbers the difference becomes less and less significant because you are taking the natural log. Maybe a mathy person can testify to that.

[@Stuartdouglas19]

the 'log' button on a calculator is base 10. So for example, 10^3 = 1000, thus the LOG of 1000 = 3. [in general 'what power of 10 is required to get a number'] 'ln' as stated in the video is to do with 'e' (the exponential) - so that's "what power of 'e' is required to get a number'

[@key2010]

god! you can see how happy Dr.Grime gets when talking about numbers i wish he was my combinatorics lecturer, would've made it alot more exciting

[@jucano]

One question related to number primes: I think that with the only number that you can form prime numbers by repeating it n times is number one : 1 and 11. There is any other combinations of the number one that get a prime number ? Thks.

[@void9720]

The first part of your statement I can tell you is true. If I was a gambler, I would guess that the answer to the question is no. I don’t know though...

[@ralaven]

@@void9720 it's trivially true as all the other n repeats can be divided by the number itself

[@thearbiter302]

Woohoo HTC One!

[@micshaz]

this is probably one of the most informative youtube comment i've ever read, lol - English is not my native language, and while i've studied various languages and speak english fluently (and have been most of my life) i didn't actually realize there was a difference between acronyms and initialisms. Virtual highfive to you.

[@xtieburn]

Just thought Id mention something. Log and Natural Log are different things, I know it says base e on the picture but it still might be confusing to people who enter log(1000000000) in to google or a calculator and get 9. (Its because its in base 10 so instead of e^9 you need to do 10^9.) You can have logs in any base, the base ten is most common in calculators and such and is appropriately called the common log. (Denoted by lg rather than the ln used in the video, maths syntax for ya.)

[@StephenSchleis]

Why aren't you The Doctor?

[@samdisum6414]

Please make a video on 1^infinity

[@erikhendrych190]

+sam disum hard time with limits?

[@LittlePeng9]

Yes, here I agree. There is also constructive argument - one can easily check that n!+2 is divisible by 2, n!+3 is divisible by 3... up to n!+n, so all n-1 numbers between n!+2 and n!+n are composite and create prime gap.

[@IAmSippycup]

I'm sure I've said this before, but I love how genuinely excited this guy gets when talking about math!

[@Tuberex]

well the title was primes are like weeds so i read math as meth

[@Booskop.]

I think we should call them Grime Numbers from now on.

[@glowstonelovepad9294]

Or call him James Prime.

[@user-gm1kn3fo7i]

If someone managed to predict the actual prime would it affect Rieehman hypothesis in any way? For instance if we know the precise 500,000,000th number and not just approximation

[@TheLetsPlayGuy98]

So I just asked my maths teacher what the PNT is... He had no clue whatsoever :) What a great teacher I have!

[@zerrickk]

By definition, according to Merriam-Webster, an acronym is "a word (as NATO, radar, or laser) formed from the initial letter or letters of each of the successive parts or major parts of a compound term." The proper term would be "abbreviation." All acronyms are abbreviations, but not all abbreviations are acronyms.

[@SethWatersVlogs]

As a musician, it's nice to have opportunities to engage in STEM disciplines in fun ways, such as this channel! Also, if Dr. Grime is single....I call DIBS!

[@shivamchauhan19]

A twin prime is a prime number that has a prime gap of two

[@quacking.duck.3243]

4:33 love you for using the long name system! :D

[@chunkyq]

1 is a special case. It is neither prime nor composite. This came up in a Numberphile video. Check out the list of prime number videos in the description.

[@fishermanWyatt-qg6tw]

But 420 isn't a prime number

[@grantmacdonald3904]

True

[@iansragingbileduct]

+Morgan Freeman But 419 and 421 are, so it all evens out

[@MrTeknotronic]

yay for twin primes

[@jwmmath]

...just add 1. presto!

[@aidan3434]

The person who made the PNT shouldn't have reused pi. He should have used CAPITAL PI -> Π

[@bolerie]

+Aidan Dorgan Π is reserved for the product over a set of terms.

[@aidan3434]

OK that makes sense

[@commandershepard1944]

Aidan Dorgan He should have used 8=====D

[@CraftQueenJr]

As opposed to π

[@CraftQueenJr]

δ maybe?

[@theSUICIDEfox]

This stuff is crazy people even thought it up. What sort of practical applications does it have?

[@ryank8843]

nf is the original formula's variable, but I was saying if you make ni in your formula 1, you don't quite get back the original. It makes sense that if nf is 1 the fraction of primes between 1 and 1 (an interval of zero) should be undefined, since it's a formula for non zero lists of integers.

[@fossil98]

Primes are like weed... Oh. By the way, it is impossible to pause a video with James and have his face look normal ;).

[@tavor29]

watched it 3 times.. didn't understand anything lol

[@TheSuperZombieNerd]

The theorem n

[@SomeMathematics]

I would suggest doing a video on e too :) It is a very important number in analysis and pops up many other places too. Maybe also say how they derived it from one of the definitions like d e^x /dx=e^x, and the equivalence of some of the definitions e.g. lim (x->inf) (1+1/x)^x = e, e=sum (from 0 to inf) 1/n!)

[@nathansmith3608]

_I'm proud of California, for legalizing primes 😎_

[@AmonAmarthFan609]

I read this as "primes are like weed" at first, and ironically I'm actually high as fuuuuhhh #nerdscanbestonerstoo

[@AmonAmarthFan609]

m4kefile there's different types of irony

[@KartonRealista2]

+m4kefile it's not an irony, but the situation is ironic.

[@elliottmcollins]

Yes, and for precisely the reason you're suggesting. There seems to be some confusion in the replies to your question, so let me clear up that two infinite sets have the same cardinality ("size") if there's a 1-to-1 mapping from the elements of one set to the elements of another. Interestingly, primes, positive intergers, intergers, and fractions are all equally sized sets.

[@ConnorOstus]

Awesome! Thanks for the info, good to know.

[@TheHortoman]

i got here by shearching 420

[@kipvis924]

Error:410 upper lips not found

[@anticorncob6]

The nth prime will always be greater than n, and the estimate for the nth prime is n*ln(n). If you check, n*ln(n) grows bigger as n gets bigger. And ln(n) > ln(m) provided that n > m. No contradictions. If you disagree, please elaborate.

[@SomeMathematics]

It depends on your definition of division. In abstract mathematics, we define every single operation. Some definitions of divisibility apply only to members of certain algebraic structures, such as a division ring or field, and infinity is not a member of these structures.

[@spinn4ntier487]

Division is the only basic function that converges instead of diverges Addition and multiplication tend towards infinity while subtraction diverges to negative infinity Division converges to 0

[@yash1152]

4:03 i like twin primes example for reminding myself: (5,7); (11,13) and so one.

[@spongebob358]

I read the title as Prime Weed (DMT) Pie for the munchies too at the start, How joyful! and then Constant E.

[@lejink]

Dr.Grime is my favorite :)

[@brian_jackson]

Correction. PNT is NOT an acronym. It is just an abbreviation. An acronym is a special abbreviation that spells a word, or is pronounced as a word. So, NATO is an acronym, because we say it like a word. So is PIN. TLA is not. It actually stands for "Three letter abbreviation".

[@skit555]

Great vid as usual but this title... Just the best one you've made. I lol'd when I read the description >_

[@HIRVIism]

They have done a video on this. James explains the long system and the short system in it.

[@FenixComputers]

if your were my math teacher, I would rush to the math class. I really like your videos keep up the good work. thumbs up for keeping me interested in math

[@islandbuoy4]

why doesn't the sound work on all of your videos for me? is it how you upload them? the sound works fine on 'some' videos I do not think it is at this end ... what are the choices for uploading?

[@suzukichopper]

+Archangel Raphael It is on your end. I had the same problem and it turned out to be the speaker setting for my surround sound. The number of speakers selected ironically enough.

[@MikeOfKorea]

I didn't mean stare into the camera, just see the camera for what it is, your audience's eyes. You can glance at the camera once in a while to acknowledge that others are watching and not just the guy who's filming.

[@KawallaBair]

Dr James Grime the King of Prime.

[@gregorscott]

Nice HTC one James!

[@burk314]

If you get deeper in mathematics (extending number theory into rings) it gets clearer. Units are the numbers with multiplicative inverses, while primes are numbers p where p dividing ab implies that p divides either a or b. The familiar definition of not having a factorization ab with a and b not units is instead called irreducible, though they are the same for integers. The point is that 1 is fundamentally different than the primes. (Note the integers actually have two units: 1 and -1)

[@shinjig]

your right, I wrote initialization when its actually initialism. Look up on google the difference between an acronym and an initialism which supports my correction of the guy saying that PNT is an acronym when its actually an initialism.

[@Vulcapyro]

(2) For example p_3 is 5 and is estimated 3*ln(3) = 3.296. The absolute error is 1.704, and the relative error is 1.704/5 = 0.3408. p_1000 is 7919 and estimated 1000*ln(1000) = 6907.755. Absolute error is 1011.245, and the relative error is ~0.1277. Notably, the relative error to the magnitude of the number is lower, but of course the number is larger and so the absolute error is also larger. It should be intuitively obvious that it gets unfeasibly hard to approximate primes as they get large.

[@diggsfather]

James Prime is back again

[@SoteriosXI]

Dear Numberphile, you're awesome! On that note, can you do more group theory and abstract algebra? :)

[@SwapnilDeshpande]

Your videos are as awesome as always! Could you also make a video about the number 'e'? Its important to both mathematicians ans computer scientists.

[@Milancius]

Ln is log for the base of e, while a log function on a calculator is a log for the base of 10. That's why ln is called natural log

[@SomeMathematics]

Hi, Pi(n) is the number of primes less than n. It grows as n grows larger, and there is no contradiction as the number of primes less than n increases as n increases. What you are thinking of, the rarity, is given by the proportion, Pi(n)/n which tends to 1/ln(n) as n tends to infinity. This becomes smaller as n, and ln n grows larger. Hope to have helped.

[@BearsOfWar1337]

I thought it said "Primes are like weed"...it perfectly matches James' face in the thumbnail

[@maximunnit]

if n=1 then the n

[@Brahmintipper]

James was rounding numbers like a physicist here.

[@SahilChaturvedi]

Nearly 45000 views and only 2k likes?? Come on, guys! Show Numberphile some love!

[@MalcolmCooks]

dr james grime always has the best thumbnails

[@ghfreakwierdo123]

The hypothesis about the gap between primes is saying that there are an infinite number of gaps between primes less than 70,000. But there are still an infinite amount of gaps that are larger than 70,000. 70,000 is by no means the largest possible gap between primes

[@MrMonishSoni]

I Like this one - Bertrand's postulate (Bertrand-Chebyshev theorem) : Chebyshev said it, but I'll say it again; There's always a prime between N & 2N. :)

[@BrianSheppard]

When you just write "log" it is usually interpreted as log base 10. "ln" is log base e also known as "natural log".