Gaps between Primes - Numberphile

An exciting paper about gaps between prime numbers - a step closer to proving the twin prime conjecture.
More links & stuff in full description below ↓↓↓
The proof was published by Yitang "Tom" Zhang from the University of New Hampshire.
We discuss it in simple terms with Numberphile regulars - physicists Ed Copeland and Tony Padilla from the University of Nottingham at • Gaps between Primes (e...
Brown papers from this video available: bit.ly/brownpapers
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[@isacharjones]

*Professor answers telephone* "Hello! Hi Babe..." Brady: "Was that Ed?" Loooool

[@naman4067]

8yr 😢

[@SquirrelASMR]

@@naman4067 what why sad did Ed die@?@?!?!?!

[@tomansager1]

@@naman4067 9yr*

[@tiletapper4ever]

​@@tomansager1 now 10. Time flies

[@JLizard]

I had Yitang Zhang as a Calculus TA at Purdue. He was full of energy and excited to teach, apologized for his English and would stay after class if anyone ever had questions. I remember him after all these years because Yitang Zhang is that special! 😁

[@nb8947]

8:00 "You'd just think there's no way you're going to find your true love. You probably wouldn't even bother going out of the house you'd just stay in an watch p-... Jeremy Kyle or something." Great save

[@non-inertialobserver946]

Lol

[@jamesmonroe3043]

@@non-inertialobserver946 Thanks for the contribution.

[@non-inertialobserver946]

@@jamesmonroe3043 No problem

[@georgetohme5658]

haahahhaaa

[@SlingerDomb]

Didn't notice that at all. LOL

[@unvergebeneid]

The progression "twin, cousin, sexy" is worrying....

[@fyggy5480]

*COUNTRY MUSIC INTENSIFIES*

[@bradleylim993]

SWEET HOME ALABAMA

[@insolubletoaster8133]

idk, seems like duck, duck, goose.

[@sagittila]

twin=2, [not sure why they picked ‘cousin’ for 4], but also sex=6

[@pranav3848]

Hey guys, I just found a new set of primes! I'm going to call it incest primes

[@tommykarrick9130]

“Oh the highest known is only in the trillions?-“ //starts writing the times 2 “oH”

[@connorlangan04]

Do you code?

[@kevinderoo3880]

@@connorlangan04 Lol I thought the same thing, since he used "//", which is a prefix for comments in coding

[@anandsuralkar2947]

Same

[@anandsuralkar2947]

@@kevinderoo3880 lol

[@Superfui]

Says "three trillions" but writes in short count.

[@OmanshuThapliyal]

The said Prof. Yitang Zhang was actually forced work in a Subway because he was unable to find academic positions after completing his doctorate from Purdue University (his adviser, Dr. T. T. Moh's role is somewhat controversial). However, he eventually gained the fame he deserved.

[@user-hm8hd2nc6u]

His doctor paper proved his mentor s paper was wrong, and his mentor lied to him about writing rec letter for him

[@wyboo2019]

this guy's kind of my new hero. i'm a butcher that can't go to college but i spend all of my free time studying mathematics (ie reading through textbooks and doing the exercises) and i hope to one day prove something

[@evandickinson3254]

That’s really interesting. I hope you keep up with it!

[@rubbersidedown7992]

Atleast stop the killings

[@themanwhoknewtoomuch6667]

Read the book "The Princeton Companion to Mathematics" probably 1000 pages which has a fantastic problem solving essays at the back about Abel Prize winners and what not.

[@rubbersidedown7992]

@@themanwhoknewtoomuch6667 surely great content.. Nice suggestion. Thanku

[@rubbersidedown7992]

@@themanwhoknewtoomuch6667seemingly great content. Nice suggestion man Thank u

[@DSAhmed]

Einstein worked as a clerk... to all you subway sandwich artists: It can happen.

[@budesmatpicu3992]

but he did not study gender shitudies or environmental managament

[@davidwright8432]

Careful! A 'clerk' who used his technical knowledge to assess patents, not a clerk who sharpens pencils and gets sandwiches for people.

[@pratheepaselvarajah1949]

@DSAhmed, S.Ramanujan worked as a clerk too! They worked just to feed themselves and to dedicate their life thinking of mathematics.

[@jeltje50]

@@budesmatpicu3992 haha gender studies bad >:(

[@jeltje50]

@@rabbitofknowledge8051 well it teaches about our culture and how we understand ourself. If gender studies is useless, then so is history or sociology.

[@numberphile]

The brown papers from this video are available - check the full video description for link. (and don't miss the extra interview footage - some nice stuff in there!)

[@HypnosisBear]

Oh thanks.

[@likeriver]

A brilliant mathematician AND a sandwich artist? damn, save some luck for the rest of us.

[@ilonnolan9259]

Squishling // Minecraft And More who cares...

[@billygarvey633]

From my state too :)

[@spiderjump]

likeriver the sandwiches he made were ok .not fantastic.

[@RedRad1990]

The ladies are all over the guy.

[@ErinSanJuan]

Tom Zhang was my math teacher in college.

[@sebster100]

Erin San Juan How was he as a professor?

[@lotrbuilders5041]

sebster100 most of my university teachers aren't actually professors. Just normal people who teach between research

[@ErinSanJuan]

Yeah, I had some professors that felt like we were a nusance in the way of research. Didn't feel like that with most of my professors though.

[@richardchamberland3381]

He was mine too!

[@BazNard]

"Was that Ed?" - brilliant

[@brianbell504]

No was aerinay... Lol

[@jayeshjagtap1296]

No this is Patrick

[@ZER0--]

7:05 When he mentioned prime numbers "falling in love", I thought, wouldn't primes "fall in love" with sexy primes the most ?

[@TheCriticsAreRaving]

+Paul L The Paul L conjecture!

[@U014B]

Also, why are twin primes and cousin primes not verboten? 😕

[@Alexagrigorieff]

Twincest!

[@YodaWhat]

@@U014B With incest, the danger is from the commonality of too-similar genes. With numbers, the least possible commonality is between prime numbers, which all share only ONE common factor. Therefore, it is indeed proper and best for primes to "date" primes.

[@pietervannes4476]

nah the sexy primes only fall in love with eachother.

[@joejoey8889]

Two years passed, I am still fucking in awe.

[@joejoey8889]

Merlyn White-Aldworth Yes, when great minds demonstrate great results, you can feel that intellectual orgasm like lasting forever.

[@StreuB1]

I could listen to these types of discussions endlessly.

[@NoriMori1992]

Fun fact, the issue of _Annals of Mathematics_ that published Zhang's proof is one of only two back issues on their website that is listed separately with its own price. The other one is the issue that published Fermat's Last Theorem.

[@joriswelles]

I read this a few days ago in a newspaper. Really amazing to see videos so accurate at the news. Keep it up!!

[@jeromechauvet6488]

Before he became famous, Zhang was a complete unknown... Waow, that's insane indeed.

[@SkyFoxTale]

AH what? The primes with a small difference between them are siblings, and they're also falling in love with each other?

[@yamahaU3]

+Sky Fox THAT'S what you got out of the video? You have a talent of missing the point.

[@SkyFoxTale]

yamahaU3 No I got the point I just thought that was funny :P It shows the dangers of analogies haha

[@astherphoenix9648]

someone had to go there innit

[@Azrage]

Probably twins because they are so similar, even without being related, that people call them twins.

[@Azrage]

You have a talent for being an asshole who misuses psychiatric terms, عمر حليله

[@MRMIKE276]

The fact that this man couldn't get a job at a university and was forced to work at Subway is a pretty big tell for me and the US school system. The guy in the video was smiling while he said it as if it was an amazing fluke but it makes me sick because it happens way too often.

[@hansschmid1298]

Mr Mike I always hoped than in the US all might all better because of a more open minded academic world. It's the US-Americans who always offered some shelter to the nerds (Gödel, Einstein, von Neumann, Carnap, Tesla ...) and who have this unique tradition of mavericks and freaks in science and arts (Feynman, Nancarrow, Riley, Pollock ... ).

[@jonaskolousek2471]

Mr Mike same thought m8

[@spiderjump]

Mr Mike he was born in shanghai and educated in China and only came to the USA in 1985 when he was 30.

[@wntu4]

It had nothing to do with the US system. His thesis poked holes in his advisers work. Said adviser did not take it well and refused to write him a recommendation.

[@metal_brrr_2005]

@@wntu4 I think his story can be made into a movie, better than A Beautifual Mind lol

[@nobodycares9797]

5:51 so... was that Ed?

[@TiKayStyle]

19. Sep. 2016 we found higher prime twins: 2996863034895 · 2^1290000 ± 1 They have nearly double decimal digits. That shows the exponential growth in calculating systems.

[@louisfox888]

how is that a prime, when it ends with the digits 4 and 6? i mean any number ending with a 5 multiplied by any number still ends up with a 5? so does this number?! i asked myself the same question about the number showed at 2:27

[@marcinukaszyk4698]

Loupe Bakoll it ends with 5 or 0 and then you add or subtract 1. Prime must be even so it has to end with 0 to get 9 or 1 at the end of prime.

[@LordOmnipraetor]

8:09 Well saved :) "you're just gonna stay in and watch ... Jeremy Kyle"

[@uplink-on-yt]

I'd rather watch what everybody else thought instead of Jeremy Kyle. Numberphile videos you pervs, what else?

[@vishalmishra3046]

There is a newer larger twin prime found - 2996863034895 x 2^1290000 + 1 (as of December 2018) - much larger than the one mentioned in this video.

[@michelelandolfi8860]

He mentioned the twin primes, not the largest prime. The number he mentions is not a number, it is a couple

[@numberphile]

I am not entirely sure the paper will solve your problem, but definitely read it!

[@fkaparkppl]

What?

[@Matthew-zy5sr]

Wow

[@clattereffect]

Zhang used the Riemann Hypothesis to come up with Pn+1 - Pn

[@genesyz]

There seemed to be some struggling with the pronunciation of Zhang's name. The Zh is pronounced like J in "Jack." The vowel is pronounced "ah," like the vowel sound in "lot." The Yi in his given name, Yitang, has the same vowel sound as the word "eat," and the tang rhymes with his family name, Zhang. So his name is pronounced like: Yee-tahng Jahng.

[@user-hm8hd2nc6u]

Not jiang, but zhang, like cheek, remove eek, remember that ch sound then change to zh sort of like drown but in zhi u own way, then combine with ang, thats zhang

[@SteveThePster]

I love these massive bounds that you sometimes get in proven mathematical theorems - e.g. this 70 million which will probably end up boiling down to N

[@MatesMonchis]

Nice prediction! The number did eventually go down to N < 246 using the techniques in the paper.

[@NoriMori1992]

"I was going to ask you to describe this in terms that you would describe it to, say, your daughter… And then I remembered your daughter does economics at Cambridge."

[@RancidPrune]

Yes, I also heard the very first sentence of this video.

[@aer9498]

It was very funny. Some people didn't understood the actual meaning. He meant: "I was going to ask you to describe this in terms that you would describe it to, say, your daughter… And then I remembered your daughter does economics at Cambridge. So I will add that you do not have to explain what a prime number is".

[@NoriMori1992]

@@RancidPrune And?

[@AlphaNumeric123]

Aer someone got rejected by Cambridge...

[@deangel9128]

.....you just quoted the first sentence of the video. What is your contribution? Did you think it was funny? What do you have to say about it?

[@numberphile]

quite possibly - it also comes up in the "extra footage" for this video!

[@highlewelt9471]

2:24 just take a moment to realize that these numbers allow *no numbers below them* to divide them

[@michawielgus9827]

*lower than their half

[@yattermansangou]

@@michawielgus9827 why their half?

[@michawielgus9827]

@@yattermansangou idk its been like a fourth of the year ago

[@michawielgus9827]

Im different person

[@yattermansangou]

@@rabbitofknowledge8051 but since they're prime, wouldn't any number below them (except 1 of course) not divide them?

[@richisnang1]

8:10 'you probably wouldnt even bother going out the house, youd just stay in and watch p... ...geremy kyle or something'

[@deangel9128]

Wow somebody stole your comment.

[@maggru91]

Been waiting for this video ever since I saw this Gaps in primes story on Wired. I just knew you'd pick it up and explain it elegantly with the help of one of your trusty professors :)

[@MrShysterme]

Intuitively, larger numbers are more likely to be divisible by some whole number (and so not be prime) because there are more numbers below them that are potential divisors.

[@EANTYcrown]

odd thing about the "cousin primes" for some reason in spanish people call prime numbers, "cousin" numbers (translation issues) so those would be cousin cousin numbers

[@ferko28]

It's not a translation issue, both words have the same semantic origin.

[@ferko28]

It's not a translation issue, both words have the same semantic origin.

[@EANTYcrown]

what i meant by translation issue is that in english prime is nothing like cousin, whereas in spanish they are the same word

[@angelmendez-rivera351]

EANTY Crown They are not the same word. Rather, the meaning of the word "primo" as in "cousin" is semantically derived from the word "primo" as in prime. The etymology is that, in medieval Hispanic culture, the theory of families was that there exists a nuclear family, and the siblings from this nuclear family is what used to be more formal called the zeroth order siblings, because the degree of separation is 0, because they are the children of your own parents, so the number of generations you have to rewind back before you have a common ancestor is 0. Then cousins are "primo hermanos" since the degree of separation is 1: you have to go back 1 generation to achieve a common ancestor: you do not share parents, but do share a set of grandparents. This makes sense since the word prime, etymologically, is directly linked to the number 1. Then you have "segundo hermanos", which in English would be the equivalent of second cousins, which would be second-order siblings, because the degree of separation is 2: sharing great-grandparentsc but not grandparents or parents. However, this terminology became completely lost after some time simply because second cousins tended to become irrelevant in the family life for practical matters, so the phrase "primo hermanos" was reduced to simply "primos". Of course, you still do see the phrase primo hermanos occasionally in books and in some parts of South America, but for the most part its obsolete, or it otherwise became corrupted

[@jimcorbin5658]

I love it when the Elite learn that someone from the out side does something they haven’t.

[@marcelorosa1973]

Tony Padilla seems to say "this kind of breakthrough should be achieved by people LIKE ME and not by non-elite harvard/mit/yale star guys"

[@AllHailZeppelin]

Fun Fact: the largest discovered twin primes are numbers of size ~10^200706. In order to write a number with 200,706 digits on an 8.5x11 sheet of paper, each digit (assuming it has equal width and height) would have to be 0.021 inches wide. That’s *roughly* the size of a small period written with pen.

[@Krish_202]

Size ? All this coming from measure theory

[@AllHailZeppelin]

I just meant if you changed the number from being expressed as a*2^b to a*10^b (ie. 10 raised to some power), the exponent would be ~200,705 (so add 1 to get the number of digits those numbers have). Point is: those pair of primes are so gigantic that even with the smallest typing or hand-writing imaginable, they wouldn’t fit on a sheet of paper.

[@Krish_202]

@AllHailZeppelin yeah you can prove this from measure theory

[@AllHailZeppelin]

@@Krish_202 prove *what*? All I was talking about was the magnitude of the numbers at 1:55….

[@Krish_202]

@AllHailZeppelin oh my bad I was reading your words as the bounds between those primes which are gigantic. Prove means those bounds are constant or not when thought of as real numbers . Like if you suppose create a circle of width suppose diameter suppose l around each prime having those bounds , then from the fact that all rational numbers have a bound of e^(epsilon + l/[2^(lambda(Q)] where epsilon is just the distance between each those circles around rational numbers and lambda(Q) is the density of rational numbers . Then from this fact you can just say that boundedness between primes is just e^(epsilon + Lp/2^[lambda(Zp)] where Zp is density of prime numbers and Lp is the gap which you will take around the primes (those circle distance) . This is what I meant from proving it in the measure theory way

[@bbsonjohn]

Talking about sad story in Maths. Lu Jiaxi in combinatoric is the sadness story. A genius basically left unseen by the world until the very last moments of his life.

[@stibar8050]

Now I know why I can't find true love, it's because I'm a big prime!

[@vighnesh153]

I don't know why but this consolidates me.

[@kasajizo8963]

Proof that the gap between two consecutive primes can be as large as possible: Let the two primes be p1, p2, and let us introduce a new variable, n. Since n! is a common multiple of every integer up to n, n!+1 divides 1, n!+2 divides 2, n!+3 divides 3, n!+4 divides 4, etc. More generally, we can say that n!+k divides k, k

[@ericmiller6056]

Your love-lorn large primes made me wonder about the following question. I know that you can always find arbitrarily large "prime deserts": for any n!, starting with n!+2, there will be n-1 sequential composite (non-prime) numbers. But, what about two prime deserts on either side of one single prime? Can each of those also be arbitrarily large? So that there is always a "lonelier" prime?

[@RuadhanG]

1:00 Einstein was a patent clerk.

[@numberphile]

no - there could be massive gaps, much bigger than 70,000,000... but there just happen to be an infinite number of primes which sit close to each other (ie: within the 70,000,000) like waiting a long time for a bus, then two can come along almost at once! :)

[@djcarter85]

You can, in fact, always get a gap between primes as big as you want! For any n, (n! + 2) is divisible by 2, (n! + 3) is divisible by 3, and so on, up to (n! + n) divisible by n. So you get n-1 consecutive numbers which aren't prime. (I'm studying maths at uni and this came up last week!)

[@matthijs122]

Seriously love this stuff.. I am consider myself smart, but not that smart. This stuff enriches and continues to inspire me.

[@andrewbrown8956]

Tom taught me Math Proofs at UNH!

[@alwinpriven2400]

who?

[@krim_z0n739]

​@@alwinpriven2400 Yitang "Tom" Zhang

[@lee155912000]

The love story at the end is what made the whole thing make sense for me.

[@keithmakan8178]

What I learned about this is that I need to be nicer to people at Subway because they might be mathematical geniuses lol

[@dancrane3807]

You wouldn't say that, if you gave the cashier $12.03 when the total you owed was $11.98, so you could get a nickle back instead of pennies. There are no pictures on the cash register to help them out there.

[@JordanMetroidManiac]

Anyone could be a mathematical genius. They’re people, too, living in the same world that you are.

[@MinusPi-p9c]

How about being nicer to people at Subway because they're people?

[@edwardjones2202]

So you're nice according to people's IQ? Feel sorry for the people round you

[@MuffinsAPlenty]

Polignac's conjecture seems to be exactly what he said. According to Wikipedia, Polignac's conjecture is "For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely many cases of two consecutive prime numbers with difference n." That seems to be what he said at 4:45, unless I am mistaken on something.

[@FoxyTheSly]

Would it be possible to do a video specifically on the number 47? It seems to have a lot of interesting properties and I'm curious to know more...

[@benjischuneman5231]

The rule at 7:20 is finally earning the name "Numberphile" in a more literal sense

[@ishandave3542]

when he said why can't we have primes that differ by 7 I was like wait woah why??? and then oh yeah obviously in less than a second its was a crazy feeling.

[@salzwasser]

I read about this in the newspaper and hoped that you bring a video about this... And here it is! Thanks alot!! Regards from switzerland

[@stefanobianchi8736]

i was waiting for the proof.. when i realised the video was ending something inside me died

[@CandidDate]

Only special people get the proof, apparently. Nobel in number theory, anyone?

[@JustForFunST]

I just finished watching The Bachelorette, and was listening to this with half an ear while doing something else, but when I heard the words "falling in love" I was like, "Wait... Didn't I move on to Numberphile? Or am I still watching The Bachelorette?!"

[@bitmaxim]

Wonderful discussion as always.

[@theradiumgirl9298]

Thanks for the information on the mating habits of prime numbers in their natural habitat, you deserve your own National Geographic channel

[@Qermaq]

Only triplet primes are 3 5 7.

[@giannislykourinas7259]

obviously, any other triplet would have one number divisible by 3

[@Bluedragon2513]

It's only because any number ending with 5 or 0 are divisible by 5

[@99bits46]

it's cuz 1+2+3+... = -1/12

[@jamesmonroe3043]

Ménage à trois Primes.

[@pietervannes4476]

@Yusril Atfan 9 + 25 = 34

[@RHawkeyed]

well, yes, that is pretty much what they said it in the video. Conjecture, "In mathematics, a conjecture is an unproven proposition that appears correct" (from wikipedia)

[@kimghanson]

Usually I get this stuff but this one has me baffled. I don't even understand what they are trying to show with the 70 million number. I don't think they explained that part well.

[@hardcoregaming07]

I think the highest difference between two primes is less than 70,000,000. (early answer ik)

[@Caesim9]

Actually, it's like this: There are Infinite many primes. Question: Are there infinite many prime pairs with a difference of 2? New result: There is a number N, so that there are infinite many prime pairs with a difference of N! And N is a number smaller than 70 million.

[@El650Jefe]

Caesim9 so is that number N the only value which will yield an infinite number of prime pairs with a difference of N? And also so if this number N isn't 2 then does that answer the question as to whether there's an infinite number of prime pairs with a difference of 2 which the answer would be no?

[@bjrnmelhus2675]

El650Jefe Q1: Are there infinite many prime numbers with a difference of 2? Answer: Yes. E.g.: (3,5) , (5,7) , (11,13) , ... etc, just recently proved by Terrence Tao. Q2: Are there infinite many prime numbers with a difference of 4 ? Answer: Most probably alltough not yet proved. Q3: Are there infinite many prime numbers with a difference of 6? Answer: Probably. Given that the generalized Elliot-Halberstam conjecture is true then 6 is the lowest number. Q3: Are there infinite many prime numbers with a difference of 70000000? Answer: No. The lowest number is far smaller than 70000000 and if 6 turns out to be the lowest number, then there exist a finite set of prime numbers that differ by 8. The ultimate challenge is: Provide a list with pair of all primes that differ by 8. I.e..: (3,11) , (5,13) , (11,19) .. and the last term is ... [ any proposals?]

[@bjrnmelhus2675]

The above is just nonsence and the twin prime conjecture is as of yet not been proven. What Zhang has proved is that there are infinitely number of prime gaps with a size less than 70000000.

[@Kasparovwannabe]

Commas are used to separate every 3 digits(not including decimals). 394,123,543.4153241 for example.

[@LeonhardEuler1]

I attended a talk from Yitang Zhang about his work about a week ago, some interesting stuff. Not terribly difficult either (although it uses some very deep results in number theory, to be sure).

[@cesarangulo1402]

The analogy of Profesor Padilla of a loveless place is awesome! cool video

[@TheAngrySnailCo]

1:07 He literally came out of nowhere.... I somehow don't buy that.... I can't say why....

[@me_hanics]

That university isn't so strong. There are even high schools having better math teachers than those professors.. (like the Fazekas in Hungary)

[@arturzathas499]

whoosh

[@geniusfollower]

"There is nothing wrong working in subway, but normally these breakthroughs are achieved by those working in elite places. And now we've got somebody who's come out of nowhere.." Srinivasa Ramanujan: *laughs*

[@aarongreenberg159]

They've been modifying this formula over the last few months, and they've gotten down to around 600

[@abhishekYadav-bx3fo]

Now it's 246

[@olayinkaanifowose5099]

abhishek Yadav now its 2, commenting from 2037

[@gplgomes]

If you take the number:G= 2*3*5*7.... N(prime)+2 , then we wil get a gap of prime with N gaps but there is a great chance that G-1 or G-3 to be a prime as well. So there isn't limit to the gaps.

[@MikeCOYS]

5:37 "Hi babe....." *continuous conversation, then hangs up* Guy with camera, "is that Ed?" ROFL

[@roylavecchia1436]

Maybe Ed is short for Edwina lol

[@Truthiness231]

I realize there is probably something simple I'm missing here, but how does one show that there is an infinite gap between two numbers (if there is one)? Like most (maybe not most here but most people in general), I have no earthly clue as to how to work with infinity properly, but I'm fairly sure I can't just write "7 - infinitely large prime = infinitely large distance" and call that a proof of anything...

[@zioscozio]

The current bound is at 246!

[@radeklew1]

A shame, since 246! is much greater than 70 million

[@themathedumacator2611]

@@radeklew1 LOL

[@youtubeofficial624]

@@radeklew1 hahahahaha

[@bobh6728]

When you started to name the largest know double primes as 3 billion, I went WHAT? A friend and I found several larger than that by programming a Monroe calculator in the late 1970’s. Then you added the times 2 to a huge number. I was thinking there for a moment that I needed to find the results that were printed on the back side of used calculator roll paper ( low budget project). What a relief that I don’t!

[@tim..indeed]

8:08 mirror.

[@drgryz]

what I started to like about mathematics, years after I finished "education", is that what's clearly seen there - mathematicians are very much like astronauts, but they can travel just on rocket of their mind, and more or less - into the mind, exploring most abstract of unknowns, bringing back constructs that will improve all ouf our lives, if put skillfully into body, metal coating of certain mechanism or into lines of code that render a lot of our lives

[@ajkelly451]

The point is that these methods are able to prove that there are an infinite number of primes separated by a specific even number. That is a very obscure result actually considering that the primes are relatively sparse.

[@GubbiGap]

70 million - yeah close enough XD

[@d5uncr]

Lina Al-Jarallah It's infinitely closer to 2 than infinity...

[@VeteranVandal]

Lina Al-Jarallah People used his result to reduce to 6 (with a conjecture) or 246 (without a conjecture)...

[@Robostate]

+Lina Al-Jarallah Patience, my friend. Patience.

[@pussiestroker]

some number bounded by 246? or did people prove that there are infinite pairs (a,b) s.t. a - b = 246 ?

[@Uejji]

Finding large prime numbers is relatively easy, because for the largest sequential prime number we know, we can find prime numbers that are around the value of that number squared.

[@heinzie5]

8:11 - he so nearly said porn there xD

[@numberphile]

You're welcome!

[@MCSre]

Zh sounds like "jr~".

[@alwinpriven2400]

is it a russian ж?

[@diegoostoja-kowalski5551]

Alwin Priven No, it's best approximation in English is j sound (voiced palatoalveolar affricate), but more retracted/retroflex/English r-like

[@Sir_Pumpington_Of_Dumpenshire]

Infinite number of pairs of primes separated by some number (2, 4, 6, 8) etc, but this number "n" is smaller than 70 million. Am I understanding this correctly?

[@Hahahahaaahaahaa]

Did you see the 'bonus footage' video? Its length is 1901, a prime that is sadly not a twin. Given the comments at the end, I would have expected them to just sit a talk for another 30 seconds.

[@inchinaxp8663]

Yitang Zhang is a badass.

[@aarondewindt]

N = a-b where b is the current prime and a is the next prime number, a is larger than b and N is therefore negative.

[@anisometropie]

2:10 I’m surprised nobody pointed that out, but the power of the largest known twin primes, it’s 2^666669 and not 2^666689 :) 3756801695685 ×2^666669 − 1

[@NoriMori1992]

+anisometropie Benihime He didn't write 2,666,689, he wrote 2^666,689. And it was multiplied by the first number, not an exponent of it.

[@anisometropie]

+NoriMori Yes indeed, I must have been really not paying attention. That’s now edited. though, the main point is still valid. primes.utm.edu/primes/page.php?id=103792 But yeah, pointing out a digit error and making an error even worse, nahh, I’m not proud.

[@NoriMori1992]

+anisometropie Benihime Aww. XD It happens to the best of us.

[@rogerdotlee]

I think that this has got to be the defining example where "IN YOUR FACE!" would be not only appropriate but almost mandatory.

[@somerandomdragon558]

Primes that differ by 7? 2 and -5.

[@WildStar2002]

Fantastic video - as always. Love your videos!

[@jamesmonroe3043]

Thank you, But I didn't make this video.

[@OrionConstellationHome]

My TMATH 308 students loved it! Thank you!

[@TheBenolds]

It is often a point of disparagement between English and American people (Rarely mathematicians, however. Maths is our language)

[@phdtobe]

Also, notice how at about 6:39, one of the British hosts used "generically" where an American would mostly likely say "generally".

[@Uejji]

If you're asking if infinity is prime, it is not. Infinity is not a number, but rather an expression that "these numbers get larger without limit."

[@slondsha]

i love how he described it with finding your true love

[@Jkauppa]

assuming y=log2(x), transform, then do the prime gap analysis, when you assume you can plot all in the log form, using the log2((a^x)*(b^y)*(c^z)*...) = xlog2(a)+ylog2(b)+zlog2(c), you get integer multiple lines per each prime base, dimensions, plot

[@Jkauppa]

you get vectors (you have a solution target at log2(k)) that corresponds to the integer powers of the "filled" composite numbers

[@MarthMadness]

What he was saying is that if we choose a really large number N. we can find a prime that is within 70 million of another prime that both are greater than N. And that N can be anything.

[@DaffyDaffyDaffy33322]

In the English speaking world, we use commas to separate groups of digits into the hundreds, thousands, millions, etc, place. We use periods to separate the whole number from the fraction. An example would be 3,712,534.67

[@diamondkev8818]

Another hint base 7 to base 9 the difference is the number of twin primes 9-7=2 so if there is infinite primes in base 9 and infinite in base 7 there will be infinite twin primes

[@15998088]

This video is why I love Numberphile so much!!!

[@fireemblem2770]

I live right near UNH. So for a math nerd like me, hearing about Zhang is pretty exciting.

[@theultimatereductionist7592]

So, did Yitang Zhang prove the weaker condition: There exists infinitely many blocks [a,a+70,000,000]: i.e. infinitely many 'a' s, which contain 2 primes. But the a's could get infinitely more "diffuse". Or, did he prove the stronger condition: ALL blocks of size 70,000,000 contain 2 primes? i.e. [70,000,000*i,70,000,000*(i+1)] contains 2 primes for all nonnegative integers i?

[@asheep7797]

obviously 70,000,001!+2 is the start of a gap of at least 70,000,000 nonprimes, so it would be the weaker conjecture.

[@DiaStarvy]

It's a major step towards understanding prime numbers.