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

[@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

[@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.

[@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

[@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!

[@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.

[@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.

[@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.

[@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!

[@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...

[@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.

[@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.

[@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.

[@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.

[@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.

[@Makajezi]

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

[@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.

[@03doug30]

Thanks for the extra footage Brady

[@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.

[@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.

[@ArisKatsaris]

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

[@Syzygyyy]

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

[@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.

[@EmyllSomar]

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

[@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.

[@iSeeJayVee]

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

[@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.

[@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).

[@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.

[@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!

[@wouterinho7]

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

[@jacolmes]

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

[@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.

[@habibaghasafari2237]

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

[@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..

[@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 ;)

[@rhaegar_ii]

loving these longer videos!

[@Engloutiee]

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

[@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.

[@numberphile]

good to hear

[@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.

[@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...)

[@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 :)

[@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...

[@MyMustacheBeckons]

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

[@TeddyRoooosevelt]

Great video. Very minor correction at 8:00 : Dr. Padilla says Aristophanes (and so do the captions) when he means Eratosthenes.

[@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.

[@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.

[@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.

[@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.

[@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

[@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

[@MrGoDuck]

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

[@stevenvh17]

No. Two and three are primes, but don't fit the pattern. Beyond 3 you have odd multiples of 3, that's (2n+1)*3. That's always an odd number, so adding or subtracting 1 gives you a composite number. If you take the even multiples (2n)*3, that's even, so adding/subtracting 2 is even again, but adding/subtracting 1 is odd. And these are the only ones remaining after we dashed out ...15-1, 15, 15+1, 18, 21-1, 21, 21+1,etc. So all the remaining numbers are n*6 +/- 1.

[@Ali2045Ali2045]

The length of this video is 1901

[@terrattenfanger5571]

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

[@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.

[@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

[@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!

[@corbilly]

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

[@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.

[@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.

[@jelowey]

Props to you, that was kind of hilarious.

[@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

[@tobysfig]

Not always. Consider the sequence 2^n, there are infinitely many terms and as n increases the terms get further and further apart. There aren't infinitely many pairs of terms that are less than x (eg. 70,000,000) apart. One really simple series, I'm sure there are many more.

[@spudd86]

Unless you're at a university or other place that has a subscription to the journal you can't read the paper sadly. If you want a link there's one on Zhang Yitang's Wikipedia page though.

[@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

[@grandexandi]

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

[@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.

[@clarekuehn4372]

Wish you would put this in the prime number playlist. Thanks!

[@gibsonrulezz]

Brady , you should definitely check the Collatz Conjecture ! Easy to present and very interesting !

[@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.

[@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.

[@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.

[@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.

[@calohtar]

Around 8:00 there’s a mistake in the captions. It’s supposed to say “sieve of Eratosthenes”. Aristophanes is the playwright

[@banjogringo]

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

[@playerthree38]

extra footage is the best part IMHO

[@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 :)

[@MrPooee]

I love these conjecture discussion videos.

[@Urqzen]

*Sees papers *Sees camera *Sees Brady Analizing is entertaining

[@PopeLando]

Just spotted a strange error by Tony at 14:45 - the Goldbach conjecture isn't one of the Millennium Problems. In fact one way you can tell this is that none of the Millennium Problems are something that could be explained to and understood by a child, like Twin Primes, Goldbach, Collatz or even Fermat. Makes me wonder if, if still unsolved in 2000, would Fermat have been included?

[@thecakeredux]

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

[@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.

[@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.

[@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.

[@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.

[@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!

[@Metagami]

Since the 70 million is an artifact of the sieve and distribution methods, could we not work down each number from the original test value to narrow the field a bit more, or have I misunderstood how these methods work?

[@unclvinny]

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

[@debug_duck]

The computer calculates and stores all numbers in binary, without converting it. The only time the computer convertes a nuber from binary to decimal (or in any other base than binary) or vice versa, is at the input and the output.

[@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.

[@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".