The Infinite Twin Prime Problem

[00:00] - [Derek] On the morning

[00:02] the journal Annals of Mathematics

[00:06] It claimed to contain a 50

[00:09] of the oldest unsolved

[00:12] A problem that great

[00:15] called unattackable.

[00:18] But this proof didn't come

[00:21] it came from an unknown.

[00:23] (bright music)

[00:25] Someone who had once spent years working

[00:27] at a subway restaurant.

[00:29] - So, they're like, okay,

[00:30] surely this isn't gonna

[00:32] You know, we'll send it to a referee.

[00:34] - [Derek] They expected to

[00:37] but they didn't.

[00:39] So, they went through it again,

[00:40] closely studying the fragile parts

[00:42] where a proof like this

[00:45] but still nothing.

[00:47] Soon they realized they were

[00:50] - Oh, damn, he did it.

[00:52] - So, how did he do It?

[00:54] What did the experts miss

[00:57] he was working on?

[00:58] He was working on a new way

[00:59] to attack one of the

[01:02] in number theory,

[01:03] the twin prime conjecture.

[01:06] The twin primes

[01:07] are prime numbers separated

[01:10] like 11 and 13 or 17 and 19.

[01:13] As you go up, the number

[01:16] and twin primes become rarer still.

[01:18] But the twin prime conjecture claims

[01:20] that there are infinitely many

[01:25] But is it true?

[01:26] Well, one way to approach this problem

[01:28] is to look at the gaps

[01:31] as you go up the number line.

[01:33] (bright music)

[01:34] Now, at first they seem chaotic,

[01:36] but if you average them

[01:41] The average gap between

[01:43] as the natural logarithm of the number N.

[01:47] So, for example, the average gap

[01:49] between primes around

[01:53] The average gap between

[01:57] Logarithms grow very slowly,

[01:59] but they do keep growing forever.

[02:01] So, as N approaches infinity,

[02:03] the average gap between

[02:07] which is not particularly encouraging

[02:09] if you expect to always

[02:12] that are just two apart.

[02:15] But if you start checking large numbers,

[02:17] then after a million you

[02:20] 1,000,037 and 1,000,039.

[02:24] Past a billion, there's

[02:28] In fact, we have found a

[02:31] as 2,996,863,034,895

[02:38] times two to the power of 1,290,000

[02:42] plus or minus one.

[02:44] That is a pair of numbers

[02:51] If you wanted to print those

[02:54] you would need around

[02:59] All of this is to say that

[03:02] we've kept finding twin primes.

[03:04] But of course, that is not how you solve

[03:06] the twin prime conjecture,

[03:07] because you cannot physically check

[03:09] all the numbers out to infinity.

[03:11] So, we need another way.

[03:13] And around 100 years ago,

[03:15] it felt like mathematicians

[03:17] with a more sophisticated method.

[03:20] In 1923, English mathematicians,

[03:22] Hardy and Littlewood figured

[03:25] how many twin primes there should be.

[03:27] To do it, they started with one

[03:29] of number theory, the

[03:32] which tells you that the odds

[03:33] of a large number near N being prime

[03:35] are roughly one over the

[03:39] So, let's do a quick example

[03:43] Say you wanna find the

[03:47] Well, you just plug that in

[03:49] and find that it has about

[03:51] chance of being prime.

[03:53] Of course, you could use the same trick

[03:54] to find the odds that 137,039 is prime,

[03:58] which is also around 8.5%.

[04:02] So, what are the odds that

[04:05] Well, you just multiply

[04:07] that gives you around a 0.7% chance.

[04:10] Now we are simplifying a little here

[04:12] because we're assuming that

[04:15] which they're not.

[04:16] But putting that to one side,

[04:18] the odds of both a large number N

[04:20] and N plus two being prime,

[04:22] is one over ln(n) times

[04:26] for large N the plus two is insignificant.

[04:29] So, we get the odds of any pair of numbers

[04:32] of size N being a twin prime

[04:34] are roughly one over ln(n) squared.

[04:38] But of course, the

[04:40] is that the odds of a pair

[04:43] as you go to larger numbers.

[04:44] So, to count all twin primes up to N,

[04:47] we add up the odds at

[04:49] In other words, we

[04:52] from the first prime

[04:55] Now Hardy and Littlewood

[04:58] to account for the fact that primes

[04:59] aren't really independent.

[05:01] But all that added is just

[05:04] So, the final expression looks like this.

[05:07] (bright music)

[05:08] Now let's plot how many twin primes,

[05:10] this predicts up to 1 trillion.

[05:12] We see that it keeps growing

[05:13] and if we fill in the actual count,

[05:16] you'll notice that it is extremely close

[05:20] to the point where by 1 trillion,

[05:21] our estimate is only off by 0.001%.

[05:26] (bright music)

[05:29] - The only issue though, is

[05:33] or what mathematicians call a heuristic.

[05:36] It can tell you roughly how

[05:39] but it can't guarantee that they never stop.

[05:42] As Terry Tao puts it, for all we know,

[05:45] there could be this fast conspiracy

[05:47] that every time a number

[05:50] it has some secret agreement

[05:52] with its neighbor N plus two saying

[05:54] you're not allowed to be prime anymore.

[05:57] So, what we need is a rigorous

[06:02] but finding such a proof

[06:03] it turns out, is extremely difficult.

[06:07] One of the first mathematicians

[06:08] to really try was a 29-year-old

[06:12] (bright music)

[06:14] Brun was working during the early years

[06:16] of the first World War,

[06:17] and with travel disrupted

[06:19] and Europe's mathematical

[06:22] much of his work happened in

[06:27] His goal was simple to try

[06:30] or count of twin primes always increases.

[06:33] And to do this, he took a 2000

[06:37] and tried to adapt it for twin primes.

[06:39] Now the normal version

[06:41] the sieve of Eratosthenes, and

[06:46] (bright music)

[06:47] Say we want to find the primes up to 100.

[06:50] First we remove one

[06:53] Then we circle the first

[06:55] and we remove all of its multiples.

[06:58] We do the same for three.

[06:59] Circle it and remove its multiples,

[07:02] (bright music)

[07:03] and then repeat that for five and seven.

[07:06] And with these four very quick operations,

[07:09] every single number that's

[07:12] is a prime number.

[07:15] Now we stop the sieve at seven,

[07:17] and we could keep going sieve by 11 or 30,

[07:21] but it turns out that we don't have to.

[07:23] So, why is that?

[07:25] Well, the next prime is 11.

[07:27] So, let's check the numbers below 100

[07:29] that are divisible by 11.

[07:31] 22 is two times 11,

[07:33] but since it's divisible by

[07:36] We also have 33, but that

[07:39] And a similar logic holds for 55 and 77.

[07:43] But what about 11 times 11?

[07:46] Well notice that is 121,

[07:51] So, every multiple of 11 below 100

[07:54] was already caught by a smaller prime.

[07:57] So, sieving by 11 would

[08:00] - So, Eratosthenes,

[08:02] has this idea that you

[08:05] the primes that are at most square root,

[08:06] the size of the number.

[08:08] - But there's also a second

[08:11] Take the number line

[08:13] and let each prime send out a

[08:16] to the size of that number.

[08:18] So, the wave from two lands

[08:21] and the wave from three

[08:24] (bright music)

[08:26] Now, notice that wherever a wave lands,

[08:29] that number has to be composite.

[08:32] But the numbers that escape

[08:34] well those have to be prime.

[08:37] And so you end up with this

[08:41] of the sieve.

[08:42] (bright music)

[08:43] And both of these methods

[08:46] to identify the primes.

[08:49] - But Brun didn't care about

[08:52] he just needed to know how many were left.

[08:55] And to do this, he counted all the numbers

[08:58] that were crossed out and then subtracted

[09:00] this from the total,

[09:01] since whatever is not

[09:05] So, let's use this to find the

[09:09] We can keep a running count.

[09:10] So, we'll start with 100.

[09:12] We're gonna throw out

[09:14] Well, that's 50 numbers left, okay?

[09:16] Then we're gonna throw

[09:18] Well, how many multiples of three

[09:19] are there less than 100.

[09:20] 100 divided by three

[09:22] It's like 33 and a third, but

[09:26] So, okay, forget that

[09:28] We do the same for the

[09:30] but now the count has gone negative.

[09:33] Well, it turns out that some

[09:36] like six for example, it's

[09:39] So, we crossed it out once when

[09:42] and again when we removed

[09:44] So, we have to add it back once.

[09:46] The same thing happens

[09:48] which is two times five,

[09:50] 14 which is two times seven,

[09:51] 15, which is three times five and so on.

[09:54] So, we add these overlaps back

[09:56] and now the count rises to 28,

[09:59] but it's still not quite right.

[10:01] - Now we've triple added

[10:04] of two times three times five,

[10:06] so that we need to subtract one more time.

[10:08] - The same thing happens for 42,

[10:10] which is two times three times seven,

[10:12] and 70, which is two

[10:15] There's a fourth triple two,

[10:17] which is 105, but that's

[10:21] So, it contributes nothing here.

[10:23] But we'll write it in to

[10:26] After the triples, we

[10:29] two times three times five

[10:32] That's also bigger than 100,

[10:34] so it contributes zero as well.

[10:36] And finally, we add back the

[10:40] and subtract one because it

[10:43] So, the count lands at 25 primes.

[10:47] This process of alternating

[10:48] between subtracting and

[10:51] is called inclusion exclusion.

[10:53] And it makes it super easy to find out

[10:54] how many primes there are

[10:58] Now this equation still looks

[11:01] but actually we can

[11:05] Now notice that every prime factor

[11:07] we added just adds an

[11:10] We've got the two that goes over here,

[11:13] the three goes over here,

[11:16] and seven goes over here.

[11:18] And if we were trying to find

[11:21] and we wanted to add 11 as a prime factor,

[11:23] it would just add one additional

[11:27] Now in general,

[11:28] if we keep the sieve

[11:31] which is less than the square root of N,

[11:33] then the count becomes approximately this.

[11:38] - But this is still just

[11:40] What Brun really wanted was

[11:43] And so this is the part

[11:47] to find a case where both

[11:51] For example, when you're sieving by five,

[11:53] you remove the numbers where

[11:56] just as before.

[11:58] But in addition, you need

[12:00] where N plus two is divisible by five.

[12:03] So, you also remove 23, 28, 33 and so on

[12:07] because in those cases, N plus

[12:13] So, this means that while

[12:16] a prime P removes about

[12:19] the twin prime sieve removes two.

[12:22] And this changes the count to this,

[12:25] where the numerator for

[12:28] ends up getting a two.

[12:31] Now, if we plot the count of

[12:34] we see that it grows roughly like N

[12:36] over the natural logarithm of N squared,

[12:39] just as for our heuristic.

[12:41] And so it seems like Brun did it,

[12:43] but unfortunately that is not the case.

[12:47] - We said that 100 divided

[12:51] but forget about that little error,

[12:52] those little errors,

[12:54] it's very difficult to

[12:56] 'cause there are so many

[12:58] Each one of them is at most

[13:00] - Take the traditional

[13:03] when we're sieving with just one prime,

[13:05] we just have an over two.

[13:07] So, one rounding error,

[13:09] with two primes, two and three,

[13:10] there are three separate rounding errors,

[13:13] but with three primes,

[13:15] there are already seven

[13:18] each for one term in

[13:22] And so you might start to

[13:25] One prime gives two to the

[13:28] two gives two to the

[13:31] and three primes gives two

[13:33] to the power three minus one terms.

[13:35] In general, if you sieve by K primes,

[13:38] you get roughly two to the

[13:42] but we can drop the one

[13:43] because it doesn't really

[13:47] So, that means that if you're

[13:49] the error grows roughly

[13:53] But if you're saving for twin

[13:56] And now the error grows roughly

[13:59] And so this causes a lot of

[14:03] (bright music)

[14:06] And then here for the twin

[14:11] The main term grows a bit more slowly,

[14:15] but the error term grows much faster.

[14:18] So, it still starts off slow.

[14:21] And so you can quickly see

[14:22] that once these error

[14:25] you get into trouble.

[14:26] So, the issue he was facing

[14:28] and the main term, you know,

[14:30] it grew in the way he wanted to,

[14:32] but then it just got overtaken

[14:34] and dominated by those error terms.

[14:35] - Exactly.

[14:36] And that's all of analysis,

[14:37] in general analytic number

[14:40] is the fight between the main term

[14:41] that you think is the truth,

[14:44] and getting the error

[14:47] - So, for twin primes,

[14:49] the main term grows roughly like this,

[14:51] something like N over the

[14:55] But that's not the true count,

[14:56] because to get the true count,

[14:57] we must add and subtract the error terms,

[15:00] which gives us this upper and lower bound.

[15:03] And now you can see the problem.

[15:05] Because the true count can

[15:08] and as you can see, a big

[15:11] So, to prove the

[15:14] is we must show that this lower bound

[15:16] is always positive and growing.

[15:19] But that is where we run into a problem

[15:21] with the sieve we've been using.

[15:23] Because the more primes we sieve by,

[15:25] the more those error

[15:29] and we just can show this.

[15:30] - And so what Brun eventually realized

[15:33] is that if you weaken the sieve,

[15:35] if you don't sieve all the

[15:37] but to up to something less than that,

[15:40] and you are very careful about

[15:44] you can actually take

[15:46] - So, run a sieve by fewer factors,

[15:48] only up to N to the power one over 10.

[15:52] And as a result,

[15:53] he gained enough control

[15:56] But this chooses a trade off.

[15:58] Say you wanted to find

[16:00] up to 10 billion,

[16:02] then using the square root method,

[16:03] you would need to sieve up to

[16:06] which is 100,000.

[16:08] But that gives you a lot of error terms.

[16:12] But with Brun's method,

[16:13] you would only need to

[16:16] to the power one over

[16:20] But there's a catch.

[16:21] Because Brun, didn't

[16:24] there were some survivors

[16:26] but numbers with many prime factors,

[16:29] in Brun's case up to nine.

[16:31] So, what he actually ended up proving

[16:34] is that there are infinitely many pairs

[16:36] of numbers two apart,

[16:37] where each number has at

[16:42] - Brun's techniques were improved

[16:45] and we went from nine prime

[16:49] to three prime factors.

[16:50] Until in 1973,

[16:52] a Chinese mathematician

[16:56] and proved that there are

[16:58] where P plus two has at

[17:02] - This is as close as you could get

[17:05] to proving the twin prime conjecture

[17:07] without actually getting there.

[17:09] We're stuck like we're

[17:12] and no further.

[17:14] - Right.

[17:14] So, that's one approximation,

[17:17] one mechanism for

[17:19] There's another.

[17:20] So, the other mechanism

[17:24] between two consecutive primes?

[17:25] So, instead of saying they differ by two

[17:27] and one of them's prime

[17:30] we're gonna try to get,

[17:32] of prime factors that it has.

[17:33] Instead we're gonna say

[17:36] but let's reduce the

[17:39] - Now remember, on average

[17:42] about the natural logarithm of N apart.

[17:45] And so the question became,

[17:48] that primes come closer

[17:52] For decades, mathematicians

[17:55] at this problem.

[17:56] By 1988, the gap had

[17:58] to roughly a quarter of the average gap.

[18:01] This means that say the average gap is 100

[18:03] at some enormous scale,

[18:05] then primes must sometimes come

[18:07] within about 25 of each other.

[18:10] But then in 2005, Goldston,

[18:14] proved a result that shocked

[18:17] - And they announced the

[18:20] 0% bounded gaps of 0% of the average.

[18:24] - Wait, what?

[18:26] - They proved that you

[18:27] as small as you want.

[18:29] This means the gap could

[18:33] or even 1000000th of the average gap.

[18:35] It could be as arbitrarily

[18:38] and infinitely often

[18:42] - When I was young, we didn't

[18:44] less than say one 10th log X

[18:46] or infinitely many primes pairs

[18:50] and Goldston and Yildirim,

[18:50] came up with a method to attack that.

[18:54] - But the method also came

[18:57] an absolute bounded

[19:01] (bright music)

[19:02] - And so that was like a big thing.

[19:04] - What was the big desire to go from 0%

[19:07] or arbitrarily small to

[19:11] - Well, because we think that the bound

[19:13] between two consecutive primes

[19:15] and right now we're showing

[19:18] - If they could get to bounded gaps,

[19:21] then they had another method

[19:25] The only problem

[19:26] was that it seemed like

[19:30] And so in 2005,

[19:32] the American Institute of

[19:35] gathering all the world's

[19:39] GPY, Andrew Granville,

[19:43] all gathered for a week in California

[19:46] with one explicit goal,

[19:48] to prove a bounded gap between primes.

[19:51] - I was a young graduate student,

[19:53] I was very lucky to get

[19:56] You're surrounded by

[19:58] on this subject.

[19:59] We spent an entire week.

[20:01] And the upshot of this week

[20:02] was basically that it's impossible.

[20:04] And Soundararajan shows

[20:07] to do this thing.

[20:08] And so as far as I was

[20:11] you know, this wasn't

[20:13] I'd have to do something else

[20:14] and I went off and and did other things.

[20:16] But there was one person

[20:19] Yitang Zhang.

[20:20] (bright music)

[20:22] - Zhang grew up in China,

[20:24] and around the age of 30 moved to the U.S.

[20:27] to get his PhD in mathematics,

[20:29] but he never got any

[20:31] and so struggled to find a job.

[20:36] He ended up living in his car for time

[20:38] and ultimately working

[20:41] including at Subway,

[20:44] and sorted receipts.

[20:47] Yet, while doing all of

[20:50] he would drive down to the local library

[20:52] to read books and

[20:56] (bright music)

[20:57] Then in 1999, he got a lucky break.

[21:01] One of his friends helped

[21:04] at the University of New Hampshire.

[21:06] So, now he could spend all his time doing

[21:09] what he loved most, math.

[21:12] Zhang said that when he was a kid,

[21:14] he "Imagined there would be a day

[21:15] that I would solve a major math problem."

[21:19] And by 2010, he had identified

[21:21] his major problem to focus on,

[21:24] bounded gaps between primes.

[21:27] But to understand what he was working on,

[21:29] we need to understand what

[21:35] See, they wanted to find two

[21:38] And to do this, they

[21:41] with some holes in it.

[21:43] So, let's do the same,

[21:44] and for the sake of this example,

[21:47] let's say the stencil

[21:49] and holes at spot zero, two and six.

[21:53] Next we place the stencil

[21:56] say starting at 12.

[21:59] And we ask a simple question,

[22:03] In this case, we see 12, 14, and 18,

[22:06] none of which are prime.

[22:07] So, we write down zero.

[22:09] Next we shift the stencil

[22:13] 13, 15, and 19, two of which are prime.

[22:17] So, we write down two.

[22:18] And then we keep doing this.

[22:20] If you slide it a bit further,

[22:21] we catch two primes again, 23 and 29.

[22:25] Now, if you keep moving the

[22:29] and it keeps catching two primes,

[22:31] then you have proven

[22:33] that infinitely often

[22:36] within a bounded gap.

[22:37] Six in this case, since that's

[22:42] But there are two problems

[22:45] The first is that we don't know exactly

[22:47] where all the primes are.

[22:49] And the second is that of

[22:52] a stencil down the number line forever

[22:54] because well, it's infinitely long.

[22:57] Fortunately, there is an easy way

[22:59] to get around that first problem.

[23:01] See, while we don't know exactly

[23:04] we do know how they behave on average.

[23:07] For example, say we take some

[23:10] from some very large number X to two X,

[23:14] then we can't predict

[23:16] in this stretch will be prime.

[23:18] But what we can do is put this

[23:21] and estimate how many

[23:25] (bright music)

[23:27] And GPY realized they

[23:30] for their stencil method

[23:32] to build an averaging machine of sorts.

[23:35] All you do is you set

[23:37] input your range, and out comes

[23:40] it caught per position.

[23:42] So, let's do a quick

[23:46] to see how this helps.

[23:48] Let's use the same stencil from before

[23:50] and use the range 20 to 40.

[23:52] Now, in reality, you would

[23:55] but for simplicity, let's stick to this.

[23:58] To find the average, you

[24:01] at the start of your range,

[24:02] and you'll look at which numbers you see,

[24:04] 20, 22 and 26.

[24:06] None of which are prime

[24:08] and so the total prime count is zero.

[24:10] Then we shift the stencil over one spot

[24:12] and see 21, 23 and 27.

[24:15] One of which is prime, so we

[24:19] Now let's keep shifting the stencil

[24:21] and keep adding all the primes we see.

[24:24] No primes, so add zero,

[24:26] two primes, add two, and so on.

[24:29] (stencil shuffling)

[24:31] Now, if you keep shifting the stencil

[24:33] all the way until the end,

[24:35] you find a total of 14 primes

[24:40] (bright music)

[24:42] And so the average is 14 over 21,

[24:45] or about 0.67 primes per location.

[24:51] Now in this case where we

[24:55] we could see that the stencil

[24:58] with two primes in them.

[25:00] But the averaging machine

[25:03] All that spits out is an average,

[25:05] and that average alone

[25:07] can't guarantee that it

[25:10] because you could just

[25:12] all of these 14 primes over

[25:17] And the same would be true

[25:21] or even one.

[25:23] I know it's unlikely,

[25:24] but it could be that every

[25:28] to bring the average to one.

[25:31] But what if the average

[25:34] What if we found 22 primes in total?

[25:37] Well, even if every position

[25:41] you would still have one prime left over

[25:43] and that needs to go in one

[25:47] So, in other words, if

[25:50] then that guarantees

[25:51] that at least one position

[25:55] And so that's the game.

[25:56] Get the average number

[25:59] but that still leaves one question.

[26:02] Even if you could do that, then

[26:06] that you keep getting bounded gaps

[26:08] all the way up to infinity?

[26:10] Well, remember you just proved

[26:15] so there's nothing stopping

[26:18] with two X, and that two X with four X,

[26:21] and then you could replace

[26:23] and the four X with eight X, and so,

[26:27] on all the way up to infinity.

[26:29] So, if you can prove that the average gap

[26:31] is above one for any large number X,

[26:35] then with this clever setup,

[26:36] it immediately extends all

[26:40] and you would've proven your bounded gap.

[26:43] (bright music)

[26:45] - Unfortunately, running the machine

[26:47] with just the current inputs,

[26:48] the stencil visits many positions

[26:50] where it catches zero primes,

[26:52] this drags the average down

[26:53] and means it will never get above one.

[26:56] So, GPY made one final

[26:59] They realize that some starting positions

[27:01] are more likely to catch

[27:04] For example, if the stencil

[27:07] it will always give zero

[27:11] So, those positions

[27:12] should not be weighted

[27:15] Similarly, if you can divide

[27:17] in the stencil by three, five, seven,

[27:19] or other small prime factors,

[27:21] it's also more unlikely you

[27:24] So, they should also be weighted less.

[27:27] Now, GPY did a few simple checks

[27:30] to assign it a weight,

[27:33] it is that this position contains primes.

[27:36] This turned their averaging machine

[27:38] into a weighted averaging machine.

[27:40] And as a result, that

[27:43] With their updated machine

[27:45] they proved that if you

[27:48] then in the range X to two X,

[27:50] you could get the weighted

[27:54] but not above one.

[27:56] They were close, but they ran into a wall.

[27:59] A wall that came from their weights,

[28:02] because to figure out how to

[28:04] they needed to know how

[28:07] in something known as an

[28:09] These are basically just series of numbers

[28:12] where they're all the

[28:15] So 3 7 11 15

[28:17] that's an arithmetic progression

[28:21] Now, GPY needed to know

[28:24] over many such arithmetic progressions

[28:26] with all kinds of different step sizes.

[28:29] Fortunately, they knew

[28:32] they could use to do exactly this.

[28:34] - It's the substitute for the

[28:37] and it turns out that that's

[28:40] in a lot of these arithmetic applications.

[28:42] - But that theorem

[28:45] It only works for step sizes

[28:48] or X to the one half.

[28:50] The technical term for this one half

[28:52] is the level of distribution or theta.

[28:55] It tells you how large the step sizes

[28:57] can be while still getting

[29:00] in those progressions.

[29:02] And mathematicians believed

[29:03] that the theorem worked

[29:06] but this was never proven,

[29:08] and that is where the trouble lies.

[29:11] When GPY ran their

[29:14] they found the maximum

[29:17] was two times theta.

[29:19] In other words, they

[29:22] but they could never actually cross it.

[29:25] - They showed in their method

[29:26] that if you could get beyond a half,

[29:28] you could do 0.5000001.

[29:31] Then instead of just getting

[29:34] they can get bounded gaps.

[29:36] The primes would be some

[29:38] - That's what the 2005

[29:41] They kept trying to push

[29:45] but they ended up concluding

[29:49] From 2010 on, Zhang spent

[29:52] at it as well, internalizing

[29:56] working day and night just to

[30:00] (bright music)

[30:02] But by the summer of 2012,

[30:04] Zhang was exhausted, and

[30:08] Hoping to clear his head, he

[30:12] where one evening while they were waiting

[30:14] to leave for concert,

[30:15] Zhang stepped outside

[30:19] looking out for the deer that

[30:23] but no deer came.

[30:26] So, he was just walking and thinking

[30:28] when all of a sudden

[30:33] He hadn't brought any notes or paper,

[30:35] but Zhang believed his idea would work.

[30:38] (bright music)

[30:41] - You see, GPY had to count primes

[30:42] in many different arithmetic progressions

[30:44] with all sorts of step sizes.

[30:46] But Zhang focused on a

[30:49] one's built only from small prime factors,

[30:52] and he realized he could

[30:55] into a form where most

[30:58] This allowed him to push

[31:00] by a tiny fraction, just one over 584.

[31:06] Zhang spent the next year

[31:09] and then on the 17th of April, 2013,

[31:11] he sent off that curious email

[31:16] - Now, the Annals gets a proof

[31:17] of the Riemann Hypothesis every other day,

[31:20] so they're like, okay, surely

[31:23] but whatever.

[31:23] You know, we'll send it to a referee.

[31:25] And they're like, well,

[31:28] let's spend a few hours this afternoon.

[31:30] We'll find a mistake.

[31:31] We'll tell the annals, you

[31:33] Here's where the the gap is.

[31:35] And they start reading it,

[31:37] and then they flip back,

[31:39] So, they can just flip through

[31:41] this will work, but wait a second,

[31:42] you're gonna get stuck here.

[31:43] And they flip five pages

[31:46] that's interesting.

[31:46] That's how you're gonna handle that.

[31:48] Okay, but then you're gonna have another,

[31:50] it's like trying to lay down a carpet

[31:53] in a room.

[31:54] You're like, okay, I know that corner,

[31:56] that corner's gonna screw you up

[31:57] even if you managed to

[31:59] And then he goes over there,

[32:01] It fits in that corner.

[32:02] Wait a second.

[32:03] And then they flip, you

[32:05] and by the end of the week,

[32:06] they've reconstructed the

[32:10] - Wow.

[32:11] (bright music)

[32:12] - Zhang's stencil had 3.5 million slots,

[32:16] spread across a span of 70 million.

[32:18] And by proving two of those

[32:22] he proved a bounded gap of 70 million.

[32:26] When the news broke,

[32:27] mathematicians were in disbelief.

[32:29] - It was basically an

[32:31] I actually thought

[32:32] when I started reading

[32:33] and started realizing

[32:36] I thought it was probably

[32:37] I knew under a pseudonym,

[32:39] trying to avoid being

[32:41] If they were wrong.

[32:42] - Of course it wasn't, Zhang was real.

[32:45] And just over a year later

[32:46] he was awarded the MacArthur Genius Grant.

[32:49] (bright music)

[32:51] - To me, this is a

[32:54] of mathematics in particular.

[32:56] I think it's one of the very few fields

[32:58] where we have a truly honest approach

[33:02] to success and what counts as success.

[33:05] He sent in an argument,

[33:08] They looked at it.

[33:09] They didn't think that

[33:10] They sat there, they gave him the time

[33:14] that the argument was due,

[33:15] and he was immediately made a

[33:17] To me, it shows that

[33:19] We're doing the right thing.

[33:21] I think the cover of

[33:24] or whatever in 2013 is

[33:27] and how he got this bounded gap

[33:28] between primes toiling

[33:31] And maybe it's because he was in isolation

[33:33] that he didn't have the group

[33:35] To know that this was not,

[33:37] we were all told it's

[33:39] - After realizing a bounded gap

[33:44] people reworked Zhang's

[33:47] Terence Tao spearheaded an

[33:50] and they sharpened the method.

[33:52] So, every month, week or day,

[33:54] that upper bound kept coming down.

[33:56] One of the attendees described

[33:59] at the time, as I remember,

[34:02] everyone was refreshing

[34:04] who had the world record now,

[34:06] and ultimately they got the number

[34:08] all the way down to 4,680.

[34:12] - Meanwhile, there's a young

[34:16] who just got his PhD at

[34:19] and other one of these analytic

[34:22] And he's working with Andrew

[34:26] And he has a completely orthogonal

[34:31] that he'd been making

[34:32] some very incremental

[34:34] - When Maynard started, his

[34:38] "I hope you won't work on

[34:41] because I'm really pretty

[34:45] But Maynard ignored the warning,

[34:47] and came up with a different

[34:51] and within just a few

[34:55] Further bring down the gap to 600.

[34:58] But his method also proved something else.

[35:01] - He can get three primes

[35:03] The bound has to change

[35:05] you wanna put in that window,

[35:06] and that his number is better.

[35:08] And the method has nothing to

[35:11] - Wait, what?

[35:13] - One half was a pure mirage.

[35:16] It was a red herring.

[35:18] - That is crazy.

[35:19] One half was not a

[35:22] See where GPY average was

[35:26] Maynard's average grew

[35:29] times the natural logarithm of K,

[35:31] where K is the number

[35:34] So, all Maynard needed was

[35:39] - You need any number greater than zero.

[35:41] When you go up to one half

[35:43] you'll get better numerics.

[35:45] But the bounded gaps you

[35:47] you know, 0.01, not 0.50101.

[35:50] Now, curiously, Terry Tao independently

[35:54] has the same approach.

[35:56] He tells Ben Green about it,

[35:57] Ben Green is meeting

[36:00] and says, "Hey, Terry's got this new idea

[36:03] that he thinks is gonna get even farther."

[36:05] And Granville, says, wait a second.

[36:06] My postdoc, Maynard, is

[36:09] We gotta get these two guys

[36:11] Terry's a Fields medalist,

[36:13] he's like a super

[36:16] whereas James is this,

[36:21] and Terry says, you take

[36:23] You know, you this,

[36:24] it's your idea, you go with it.

[36:26] - By early 2014, Maynard joined forces

[36:29] with Tao's Polymath group.

[36:31] - It was clear that somehow 600

[36:34] and the same methods would

[36:37] but there were extra ideas

[36:40] to squeeze everything out.

[36:42] And so the current world record

[36:43] is that there's infinitely

[36:45] that differ by no more than 246.

[36:49] (bright music)

[36:49] - And that for now is where it stops.

[36:52] In 2022, James Maynard was

[36:56] Mathematics, highest honor

[37:00] - Yeah, I think that's the basic outline

[37:03] is the kind of two

[37:06] to twin primes Chen's direction,

[37:08] Brun, Chen, whatever,

[37:12] and this story that the

[37:15] so he didn't know it was impossible.

[37:16] - It reminded me of the four minute mile

[37:17] where everyone thought it was impossible.

[37:19] No one did it.

[37:21] - And then for the first time ever,

[37:22] Roger Bannister broke it in 1954,

[37:25] and after knowing it was possible,

[37:27] just 46 days later,

[37:29] another runner called John

[37:32] And by the end of 1956,

[37:37] all from knowing that it was possible.

[37:41] So, what about pushing

[37:44] Is that possible?

[37:46] Well, mathematicians have found ways

[37:48] to bring it down even more,

[37:50] but all of these results are conditional.

[37:52] For example, there is the

[37:56] which assumes primes

[37:57] are spread evenly across

[38:00] or that the level of distribution

[38:02] can be taken as large as one.

[38:04] And in 2013, Maynard

[38:08] this conjecture is true,

[38:09] then the gap plummets to just 12.

[38:12] A year later,

[38:12] the Polymath Group

[38:15] an even stronger version

[38:17] the gap drops all the way down to six.

[38:21] But without assuming

[38:27] Let me ask you, do you

[38:30] the twin Prime conjecture?

[38:31] - I am totally convinced that

[38:36] the twin prime conjecture.

[38:37] Often when I'm asked about some

[38:40] like Twin Primes or Riemann

[38:45] one way of kind of avoiding the question

[38:48] is that if you imagine you

[38:52] you'd expect to be sort of

[38:57] through when the

[38:59] So, an actual guess, if you

[39:02] is to guess that a problem will be open

[39:05] for as long as it has been open for already.

[39:07] But obviously this doesn't

[39:09] 'cause we don't know whether it's-

[39:11] - Right.

[39:12] (all laughing)

[39:13] - 125 years old or whether

[39:16] So, I think it's a fools game to guess,

[39:18] but it clearly needs a really big idea,

[39:21] but maybe it only needs one big idea.

[39:24] - Although maybe, just maybe,

[39:27] not knowing this with

[39:30] Because if we knew for sure

[39:34] then we would've likely missed out

[39:36] on most of these inventions

[39:37] and new methods over the past century.

[39:40] So, sometimes it pays

[39:46] - You know, before I started this channel,

[39:48] I was a teacher at a tutoring company,

[39:50] and honestly, it was the best job

[39:52] I'd had up until that point.

[39:54] I could be the person who gave my students

[39:56] that one big idea that

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