Gaps between Primes (extra footage) - Numberphile

[@andrewxc1335]

Prof. Zhang was my basic proofs teacher. I ended up missing a lot of classes due to mono. I regret not working with him more. Very unsurprising, though, that his paper is "crystal clear."

[@solderbuff]

What is mono?

[@rngwrldngnr]

@@solderbuff it's a disease people often get in high school and college. It's usually unpleasant but not threatening, though it can be quite a bit more serious if you're unlucky.

[@valor36az]

Infectious mononucleosis

[@sillysausage4549]

@@solderbuff in most of the English speaking world it is known as glandular fever. Americans, of course, have their own way of doing things.

[@tgwnn]

Apparently the limit is now at 246 (down from 70 million).

[@TykoBrian7]

+tgwnn source, plz?

[@tgwnn]

+Tyko Brian I wrote this a while ago but I see Wikipedia lists 246, with possibly 6 or 12! Sorry I can't look into the exact citations now.

[@siekensou77]

i thought numberphile said 16 is the current limit

[@tgwnn]

siekensou77 in this video?

[@jpchevron]

+siekensou77 16 was conjecture as well.

[@MoosesValley]

Really appreciate this extra footage. It's astounding to me that even on the edge of infinity the largest gap between primes can be bounded. 70 million seemd such a small gap (in an infinite number system). But as others have reported here, and this quote from inyen1 "Terence Tao and later joined by James Maynard who had found additional new methods, they got the bound down from 70,000,000 to 246 (and if the Elliott-Halberstam conjecture is true down to 16)." For me, 246 is just such an unexpected and insanely small gap between massive primes on the edge of infinity. Incredible work by Dr Yitang "Tom" Zhang and others who have lowered the gap and followed on from his work.

[@nicolageorgiev4350]

The gaps between primes don't have an upper bound. It's just that it's finally been proven that there exist an infinite number of primes that differ by a certain amount (in this case 70 mil). This sort of thing had never been proven before. There still will be primes that differ by more than that, but now we have a proof that for a gap of specifically 70 mil, there are infinitely many.

[@ralphy1054]

@@nicolageorgiev4350 Isn't the proof that there *cannot* be an infinite number of primes with a certain gap that is *over* 70000000? And that it is only *possible* for there to be an infinite number of primes a certain distance apart if that distance is less than 70 million.

[@lPlanetarizado]

@@ralphy1054 technically, they proved that the minimum (thats what the "inf" in the paper accounts for) gap size of 2 consecutive primes that repeats to infinity is less than 7*10^7 , its believed to be equal to 2 (twin numbers)....if by any chance, someone probe the minimum gap size is over 3 for example, that will mean that there arent infinite twin primes there is also a theorem about the maximum gap size, it involves a bunch of log functions, but its not a specific number like 7*10^7 (you will see a "sup" instead of a "inf")

[@Neme112]

@@ralphy1054 I don't think that's what the proof is saying. From what I know, there are an infinite number of prime pairs who are any arbitrarily large distance apart, so larger than 70 million too. It's just that there's an infinite number of pairs with a low distance apart as well.

[@grandexandi]

Damn, Brady. I'm translating the subtitles into Portuguese, but the word for 'prime' is the same word for 'cousin', so I get to this 'cousin prime' thing and I'm like "os primos primos"... LOL

[@raybroomall8383]

Prime cousin translates to primo nobre. Cousin prime translates to primo primo. I don't know which would be grammatically correct.

[@rogeriojunior9459]

6 years later

[@josegarci3533]

In Spanish it’s the same 🤣

[@branominal]

Massive props to the editor for realising it was a paper worth reviewing quickly

[@gyorgy11]

Eratosthenes not Aristophanes, he was a playwright. The Sieve of Eratosthenes is a miracle of elegance.

[@feliciabarker9210]

This and the Goldbach paper coming out so close together leads me to propose the twin papers conjecture

[@chattyw87]

Should've added 2 seconds to the video to get 19:01 1901 is a prime or added 12 seconds to get 1151 seconds total which is a twin prime with 1153.

[@Maddolis]

I find it hard enough to wrap my head around any number over 10000 being a prime number. This is just mind-blowing.

[@natejc93]

At UNH, I was taught by Zhang, he is a funny dude. No one pronounces his name correctly so he said at the very first day of class that is name is Tom (affectionately). I took multi-d calc with him and it was easy:P I mean its unh ya know?

[@l.z.7320]

Nate Cordova so there are no chinese people there?

[@franzlyonheart4362]

@@l.z.7320 , no, there aren't.

[@fournya]

As soon as you said "You know that's what I would have done," I looked down to see the length of the video. I love your attention to detail!

[@L0j1k]

Can we get a follow-up? I would *love* to hear how this proof has been evolved by mathematicians!

[@Einyen]

The polymath project with Terence Tao and later joined by James Maynard who had found additional new methods, they got the bound down from 70,000,000 to 246 (and if the Elliott-Halberstam conjecture is true down to 16). This was back in 2014. According to Terence Tao they could not get any further with those methods.

[@NoriMori1992]

I love how this video is a prime number of minutes long.

[@WalterKingstone]

+NoriMori Twin prime number of minutes (17 and 19)

[@NoriMori1992]

Walter Kingstone Oooh!

[@putinstea]

No it's 18:59. Which is 18.9833333333333333 minutes. Not even a whole number.

[@ZeHoSmusician]

18m59s is 18*60+59=1139 seconds and 1139=17*67...so not really prime...

[@xIPatchy]

or 1859, which is divisble by 11 to get my favorite number 169...

[@beeble2003]

GPY was mis-stated in the video. What Goldston, Pintz and Yildirim actually proved is that, for all ε>0, there are primes P>Q with P-Q < ε*log(P). Choosing ε to be very small doesn't guarantee that the difference between P and Q is small in absolute terms; it's just small compared to log(P). So, for example, if you choose ε=0.0001, it might be that the value of P you end up with is something like 10^1000000 and the only guarantee you get then is that P-Q < 100.

[@mixalakism]

The ancient Greek mathematician which had the sieve idea was Eratoshenis not Aristophanis. Aristophanis was an ancient Greek comic playwright.

[@alahijani]

+Michail Panagopoulos Actually it's Eratosthenes.

[@dimitriscollier9918]

Ali Lahijani Actually Greek people pronounce i as e :)

[@michaelempeigne3519]

+Ali Lahijani That's the right spelling.

[@DevonParsons697]

Yes it is. Consider that there are 5 numbers between two multiples of six (for instance, 7 8 9 10 11 are the five numbers between 6 and 12). Of those, two are divisible by 2 and one other is divisible by 3. The only possible remaining numbers that have a chance to be prime are the ones bordering the multiples of 6.

[@PTNLemay]

Thanks for taking the extra time to explain this further. I had to watch the first video a few times before I finally grasped what it was trying to say. And now that I do get it, I agree it's pretty cool.

[@wreckim]

What appeals to me about a subject like this, like Fermat's Last Theorem etc...is that it is so easy to understand the problem, and even picture how difficult it is to prove it--yet I KNOW I can't keep up with the math, but I imagine I can with proper explanation. It's a lot like watching a chess analysis of a a Carlsen in the World Championships...I imagine I too would make that move. Dreamers...I am one.

[@heyandy889]

Brady is really nerding out in the beginning. Usually he lets the talent do the talking, but you can tell he is a worthy candidate for the title of Numberphile.

[@greg55666]

I haven't read this thread carefully, but from what I see my impression is that I completely agree with you. An apology does not even require the word "sorry." The form of an apology is: (1) I recognize that what I did was wrong; (2) I recognize you were hurt; (3) I feel bad about it; and (4) I will try never to do it again. Saying "sorry" is, as you say, the opposite of saying sorry.

[@InfamousBLT]

Brady, do you always have this much extra footage? This stuff is fascinating...you should publish extra footage more often. It's great stuff!

[@Makajezi]

i absolutely loved what he said at the end of the video.

[@beeble2003]

Sorry for the confusion -- typo. Both of them should have been "Primes P>Q with (P-Q) < ε*log(Q). Thanks for pointing that out.

[@anxez]

Something interesting I have noticed noticed: Base six behaves pretty excellently with primes. It can really quickly be shown (and I imagine proved) that all primes end with a 5 or a 1 in base six.

[@geerin.]

They do! It doesn't show up in subboxes but you can see it via annotation at the end of the video it's usually paired with.

[@Psqad]

Those last words explains it all: original vid is lenght 859(twin prime with 857). This vid is also prime and also special: 1901 is a Sophie Germain prime(2p + 1), since 2 x 1901 + 1 = 3803 is also prime. Nice touch.

[@MuffinsAPlenty]

It depends on what your definition of prime is. In many abstract algebra classes, an element p of a ring is prime if whenever p divides ab, then p divides a or p divides b. Under this definition, there are infinitely many negative prime integers. In fact, if p is a positive integer prime, then -p is a prime. But again, it depends on the definition, what branch of math you're working in, and if it even matters concerning the problem you're working on. Most people would say no, though.

[@EmyllSomar]

The spark can also go into it. It also goes into the first one. Genius!

[@kevinohare9216]

Watching this 10 years later, it seems that the proposed proof of the weakened Goldbach Conjecture, which Tony mentions around 14:48, still hasn't been confirmed. Bummer.

[@lnkognitoX]

one guy took a right way.. figuring exceptions, not only acseptions.. worked on primes 2 weeks alredy, and I feel like gone further than any man before..

[@beeble2003]

At around 7:40, Brady expresses surprise that the bound, 70,000,000 is a nice round number. The video explains where the number came from but not why it's so round. The reason is that, once you know that the actual number isn't very interesting, you tend to just prove that it's smaller than some round number. He presumably knew that the exact bound from his proof was much, much bigger than 2 and did a back-of-the-envelope calculation to say it was less than 70 million.

[@Syzygyyy]

8:59 and 857 are twin primes, well done Brady.

[@03doug30]

Thanks for the extra footage Brady

[@EebstertheGreat]

This is true for all primes greater than 3, and it is easy to see why. All numbers are either a multiple of six (in which case they are not prime), a multiple of six plus or minus three (in which case they are a multiple of three), a multiple of six plus or minus two (in which case they are a multiple of two), or a multiple of six plus or minus one (in which case they may or may not be prime). 2 and 3 are exceptions because they are the only multiples of 2 and 3, respectively, which are prime.

[@jrhperu]

Harald Andrés Helfgott was born in Peru, he finished high school in Lima, at the Alexander von Humbolt College and is now workigh at the Ecole Normale Supérieure, in France.

[@zeru2150]

The property of being prime or not is universal for all bases. You simply factorize a number and than translate those factors into other bases. Example: 15= 3*5 in base 10 F=3*5 in base 16 17 = 3*5 in base 8 1111 = 11*101 in base 2 So if you want to work on prime numbers, you can simply pick your favourite base ;)

[@beeble2003]

The statement of the Goldston-Pintz-Yildirim theorem given in the video at 6:00 (for any ε>0, there are primes separated by less than ε*log(N) for large enough N) is incorrect. That statement is trivially true: for any ε, just take N=e^(1+1/ε). Now, 2 and 3 are primes separated by less than ε*log(N), since ε*log(N)>1. Goldston, Piltz and Yildirim actually showed that, for all eps, there are primes P>Q with (P-Q) < ε*log(Q).

[@hssgsng]

A better explanation: All numbers 6k, 6k + 2, 6k + 4 are even, if k is an integer. 6k and 6k + 3 are multiplies of 3, so all primes must be of the form: 6k + 1 or 6k + 5 = 6(k+1) - 1 = 6k' - 1.

[@terrattenfanger5571]

ive done this "takin' it back/Im sry thing" twice between december 2009 and may 2010... it is not that unheard of.

[@SpySappingMyKeyboard]

Every number in b6 that ends in 0, 2, or 4 will be a multiple of 2. Every number in b6 that ends in 3 will be a multiple of 3. I can't imagine converting to base 6 and back is the most efficient way of weeding out multiples of 2 and 3.

[@Knooblet]

For one: very large primes are used in the encryption of valuable information. Math is also very useful for programming, especially for video games (very large matrix multiplications) to determine has light reactions with objects. This channel mostly explains different ideas within Math, but the applications of these ideas are enormous. You seem like a smart individual, so figure some more out!

[@ArisKatsaris]

8:03 A guy named Aristophanes? I think you mean Eratosthenes. Aristophanes was an author of comedic theatrical plays.

[@unvergebeneid]

This is pretty amazing. When I first read the comment you replied to, I thougth "this can't possibly be true", then I checked the first couple of primes by hand, then the primes between 5 and 10000000000 with a Python script. Pretty amazing.

[@Martykun36]

59 is also a very particular kind of prime number, one that sees a fellow prime two units above, and that sees 57 two units below which is a honorary prime number

[@wshun]

According to the video, GPY states that: for any epsilon(e), u can get two numbers smaller than N with a gap in between is smaller than eN, provided that N is large enough. so even if e is very small, u may need N to be very large for GPY to work, but then the gap eN may not be small

[@AmiduTV]

You should do a video explaining the relationship between number of episodes and bradys %camera time and extrapolate to find the date from which all numberphile videos will just be brady being a boss

[@razorborne]

the primes exist independent of bases. while they may look different in different bases, they always represent the same number. for instance, 19 in decimal and 17 in dozenal are both prime, and both represent the same amount of objects. it's just a notational difference.

[@Okiesmokie]

It's not saying that 70,000,000 is the highest gap between primes, it's only saying that there are an infinite amount of primes where the gap between two primes is

[@MyMustacheBeckons]

I love that the video length (if read as 1901) is prime

[@Dayumhesgood]

The twin prime conjecture is to do with the idea that primes which differ by only 2 will always pop up, despite the fact that primes tend to be further apart as their value grows.

[@DiaStarvy]

Essentially, but there are also the "trivial zeros": zeta(z) = 0 when z is any negative even integer. The Riemann hypothesis states that the non-trivial zeros all have real part 1/2.

[@iSeeJayVee]

Really awesome video. Enjoyed this extended talk quite a bit.

[@Vulcapyro]

Not 2 and 3, but yes. Say we have a number n>3, and we put it in the form n = 6a + b where 0

[@numberphile]

good to hear

[@TessaBain]

All I can think is what's the point. To be honest, thanks to Asperger's I'm technically a genius, Particularly if I care about something, because then I become obsessed and being able to notice patterns like crazy only helps further it. One of the first things I became obsessed with and still am to this day was writing, be it numbers or letters. Math appropriately became my first whiz subjects. I not only learned what was taught but went further, even creating my own rules to do even the...

[@DiaStarvy]

Because prime numbers have an additional clause: they have to be greater than 1. Semi-sarcastic explanations aside, many of earlier theorems involving prime numbers repeatedly found themselves having to state "X is true for all prime numbers excluding 1", so it was easier for everyone if prime numbers excluded 1. One such theorem is no less than the fundamental theorem of arithmetic.

[@edderiofer]

Edderiofer's Prime Conjecture states that: If n is an element of N (set of natural numbers), then n+1 MAY BE PRIME. Not all such numbers are prime, but we can find primes with this formula as well!

[@stevenvh17]

The fundamental theorem of arithmetic says that every number can be decomposed into prime factors only in one way. For instance 252 = 2 * 2 * 7 * 9. You can shuffle the terms, but they will always be these 4. Now if you define 1 as prime, then you can do 6 = 2 * 3 = 1 * 2 * 3 = 1 * 1 * 2 * 3, so the factorization isn't unique anymore. That's why 1 is not prime. It's just a convention, so that the theorem works.

[@Rafaelfl2]

Yes, and they're the same.Remember that the greek (such as Aristotele, who the professor was talking about) didn't use our number system. They did basically all of their math based in geometry.

[@habibaghasafari2237]

thanks Brady, this is a really exciting video about primes which I didn't know before.

[@rhaegar_ii]

loving these longer videos!

[@wouterinho7]

The video is just 1141 seconds long which is a prime number. Nice work Brady!

[@MrGoDuck]

"So this has applications beyond number theory" as a Physics major those were the magic words I was waiting for.

[@SamOliver4]

Working in other bases has proven useful for CHECKING primes, on the other hand, because the divisibility rules for numbers are in fact different in different bases. For example, numbers that end in 1 in base 6 tend to be prime because those numbers can't be divisible by 2 or 3. Obviously not all numbers that end in 1 in base 6 are prime, but since most composite numbers are divisible by 2 or 3, this particular check makes it easier to find many primes in quick succession.

[@jacolmes]

Brady, I think you have the most awesome job in the world.

[@eirh]

Wow people seem to really have missed your point. You are actually correct, you can not write 2 as the sum of 2 primes. The Goldbach Conjecture actually states, that every even number GREATER then 2 can be written as the sum of 2 primes, they forgot to mention this in the video.

[@EmyllSomar]

"I can easily say without a doubt..." Made my damn day. Hahaha. Best response ever. Showing clearly that you're not only a reasonable person, capable of acknowledging your errors/mistakes, but also honest and sincere not to get your undies all in a bunch about trivial things. I am the original 'idiot' and I now accept your apology further. You have become one of my favorite KZbin commenters. LOL. Cheers.

[@prshrf]

Most scientific articles published on journals can only be viewed if you pay for the articles or subscribing to some online databases. Most universities and colleges pay these subscription fees so that students can access them for free.

[@seiterarch]

I know that was sarcastic, but I'll add mine anyway because I think it's interesting :P Seiterarch's prime conjecture (easily provable): If n cannot be described by n = m(6k +- 1) +- k where m, k are integers, then {6n-1, 6n+1} are twin primes. If you require (6k+-1) to be prime, this gives you exactly the set of all twin primes. Either way, I'm fairly sure it's not very useful.

[@Vulcapyro]

Divisibility is independent of a number's representation. You can show results about how primes look in other bases, but it's more likely that you'll get more information about working in that base, rather than information on the primes.

[@Ali2045Ali2045]

The length of this video is 1901

[@grandexandi]

And don't even get me started on 'sexy primes'. It's just 'sexy cousin'. I mean, really.

[@bhrisvfyapn]

I think it goes like this: AVERAGE gap between two primes (P and P+1) = ln(P) ln(P)=70,000,000 => P= e^(70,000,000) So, for primes larger than P=e^(70,000,000) there should be plenty pairs with larger gaps . In other words your first choice :)

[@thecakeredux]

Seeing people being excited about math is equally great as the matter itself.

[@Kumurajiva]

honly cow, who would have thought of such a thing. I would never have thought myself into that corner

[@unclvinny]

This video is 18 minutes and 59 seconds, so you got a "59" in there, Brady!

[@beeble2003]

They'll dream up new questions. But it'll probably be difficult to even understand the problem, which would be a shame. The great thing about Fermat's last theorem, twin primes and Goldbach is that the problems are really easy to understand and children often get interested in mathematics by trying to solve them. Having said that, even today, you or I can say "Fermat? How hard can that be?" and try to solve it, even though it's already done. We can look for a simpler solution than Wiles's.

[@piranha031091]

And 19 is a twin prime with 17. I think Brady initially intended to make the video 19:00 long, that 01 probably appeared after the upload.

[@alexmcgaw]

It's not that *every* prime has a twin, it's that there are an infinite number of twin primes. So, we can easily find low pairs like (3,5), (17,19), (149,151) etc but the conjecture says that there are an infinite number of these pairs :)

[@Engloutiee]

Would be really nice to hear about the Goldbach conjecture and Riemann hypothesis, and maybe even the abc conjecture, as well!

[@piranha031091]

A lot of domains use these, especially cryptography. Every secure data exchange uses prime numbers in the encryption process (for instance, logging in a website, money transfers with ATMs or between banks, VPN protocols, etc...)

[@Qubrof]

Well, my understanding is that there are an infinite number of primes with a gap of 70m or less, not that every prime is separated by a gap of 70m or less. So the next prime after the largest known prime might still be 300 quadrillion numbers away.

[@JureRatkovic]

Yes there are, and they are the same in all bases. For example 25 hex is a prime, and is not divisible by 5 obviously. Primalty is an intrinsic property of a number, it is not dependent on the way you write it. And number bases are just that, diffirent ways to write numbers :)

[@Maddolis]

Furthermore, I believe just yesterday they confirmed "reducing the bound to under 5 million," as I just read on wikipedia. That's fairly substantial!

[@razorborne]

it's just fun, man. it's like making a video about pi 3:14. yeah, that's not really pi because time isn't represented decimally, but it's still neat.

[@playerthree38]

extra footage is the best part IMHO

[@beeble2003]

You can't necessarily extrapolate from small numbers to big ones. To take a slightly silly example, every number you've ever seen written down has less than a million digits. So you could imagine that every number has less than a million digits but that's obviously not true. For a more subtle example, N^2+N+41 is prime for N=0, 1, 2, ... so you might imagine it's prime for all N. But, actually, it's only prime for N

[@MrPooee]

I love these conjecture discussion videos.

[@EmyllSomar]

They're all a big happy family of little primes and odd numbers.

[@xlrv1]

For most English speakers, "separated by two" means having two other items between the pair in question - but you don't mean that, obviously. It would be better to say "two apart".

[@robuso]

Talking about computations that require an infinite amount of time reminds me of a good topic: get your physicists to talk about Supertasks!

[@banjogringo]

Brady thanks for this big extra footage, I really enjoyed watching it!

[@sth128]

As primes get larger, the average gap between two primes get larger too. The twin prime conjecture states that despite this trend, there are infinite pairs of "twins" which means there's no bound beyond which twins don't exist.

[@JayCee235]

The universe actually covers up the number 23 as mathematicians search for twin primes, because if they ever realize that 23 isn't a twin prime they'll uncover some very dark secrets.

[@corbilly]

Nice trick, Brady, the length of this video is 19:01. 19 is a twin prime

[@l47l]

this has also a big implication of finding the NEXT biggest prime number because we can check the largest prime we know and check + =< 70,000,000 next numbers to find the next!

[@spapanik21]

No, It just means that if you get all the prime numbers above any x (no matter how big this x is), at least two consecutive of them, they differ by less than 70 mil. There is a very nice way proof that: there is no number K such that in every K consecutive numbers there is a prime.

[@tarcal87]

Interesting that you even go there, I give you thumbs up for that. I used to think that considering other bases is something special - it is not, and it is due to the limitations of our brains. other bases are only useful for small numbers - as we are talking about infinity, it is pretty much useless. And "even" and "odd" are defined as divisible by two or not, so it is free of bases.

[@davidwilkie9551]

"An order of magnitude in the way people think", or maybe it's all over long ago and we need a refresher. Thinking about Time as a substance, something like a drumhead for example, means that the numbers can be imagined as wave phenomena in Principle and actual perspective. So the trick is to think of primes as dominant probability integration positioning, ie wave-packages, and twin primes or any other, are nodal dominance/interference of symmetrical reflection, e-Pi-i numberness in time duration timing modulation. I started out with this idea from a text book on Chemical Bonding explained through QM, Phys-Chem, and accumulated observation of excellent presentations like these of Brady's. Sieves of varying grid sizes are somewhat similar to "Ring down" interference patterns. (Some like to infer Entropy, which is a lot like Temperature, and other vaguely applied words, but the real number properties are manifestations of e-Pi-i resonance, so every word description is another aspect of infinity.., loosly, or dualisticly connected, reminiscent of the solid-fluid dualism of all QM temporal substance)

[@enlongchiou]

use quadratic equation x^2-dx -n=0 as sieve can comb any gap d, have infinity solution for every gap d by induction, add up all of d prove goldbach conjecture, twin prime conjecture is it's special case at d=2, for example : a*b=a*(a-d)=n=5*3=15, (2^2+4*15)^0.5=8, (8+2)/2=5, (8-2)/2=3 two solution, if a or b is composed number have more than two solution and gap d not equal to 2, 7*5=35, (12+2)/2=7, (12-2)/2=5, 11*13=143, ((11+13)+2)/2=13, ((11+13)-2)/2=11, 7*3=21, ((7+3)+4)/2=7, ((7+3)-4)/2=3 for d=4, 13*7=91, ((13+7)+6)/2=13, ((13+7)-6)/2=7 , d=6, for prove Riemann Hypothesis use realization of sieve of Eratosthenes ,mean keep remainder, for example : pi(2^2)=4*(2-1)/2+0/2+1-1=2, pi(3^2)=9*(2-1)*(3-1)/(2*3)+1/2-3/6+0/3+2-1=4, pi(5^2)= 25*(1*2*4/2*3*5)+1/2-1/6-5/10+25/30+1/3-10/15+0/5+3-1=9.