[0:00] - [Derek] On the morning
of the 17th of April, 2013,
[0:02] the journal Annals of Mathematics
received a curious email.
[0:06] It claimed to contain a 50
page proof relating to one
[0:09] of the oldest unsolved
problems in mathematics.
[0:12] A problem that great
mathematician Edmund Landau
[0:15] called unattackable.
[0:18] But this proof didn't come
from a famous professor,
[0:21] it came from an unknown.
[0:23] (bright music)
[0:25] Someone who had once spent years working
[0:27] at a subway restaurant.
[0:29] - So, they're like, okay,
[0:30] surely this isn't gonna
work out, but whatever.
[0:32] You know, we'll send it to a referee.
[0:34] - [Derek] They expected to
find a mistake in an afternoon,
[0:37] but they didn't.
[0:39] So, they went through it again,
[0:40] closely studying the fragile parts
[0:42] where a proof like this
normally falls apart,
[0:45] but still nothing.
[0:47] Soon they realized they were
witnessing a breakthrough.
[0:50] - Oh, damn, he did it.
[0:52] - So, how did he do It?
[0:54] What did the experts miss
and what was the problem
[0:57] he was working on?
[0:58] He was working on a new way
[0:59] to attack one of the
hardest unsolved problems
[1:02] in number theory,
[1:03] the twin prime conjecture.
[1:06] The twin primes
[1:07] are prime numbers separated
by just one number,
[1:10] like 11 and 13 or 17 and 19.
[1:13] As you go up, the number
line, primes become rarer
[1:16] and twin primes become rarer still.
[1:18] But the twin prime conjecture claims
[1:20] that there are infinitely many
of them you never run out.
[1:25] But is it true?
[1:26] Well, one way to approach this problem
[1:28] is to look at the gaps
between consecutive primes
[1:31] as you go up the number line.
[1:33] (bright music)
[1:34] Now, at first they seem chaotic,
[1:36] but if you average them
out, a clear trend emerges.
[1:41] The average gap between
two primes grows roughly
[1:43] as the natural logarithm of the number N.
[1:47] So, for example, the average gap
[1:49] between primes around
100, is approximately 4.6.
[1:53] The average gap between
primes around 1000 is 6.9.
[1:57] Logarithms grow very slowly,
[1:59] but they do keep growing forever.
[2:01] So, as N approaches infinity,
[2:03] the average gap between
primes also goes to infinity,
[2:07] which is not particularly encouraging
[2:09] if you expect to always
be able to find primes
[2:12] that are just two apart.
[2:15] But if you start checking large numbers,
[2:17] then after a million you
quickly find the twin primes,
[2:20] 1,000,037 and 1,000,039.
[2:24] Past a billion, there's
1,000,000,007 and 1,000,000,009.
[2:28] In fact, we have found a
twin prime that's as large
[2:31] as 2,996,863,034,895
[2:38] times two to the power of 1,290,000
[2:42] plus or minus one.
[2:44] That is a pair of numbers
each 388,342 digits long.
[2:51] If you wanted to print those
in a book for some reason,
[2:54] you would need around
260 pages per number.
[2:59] All of this is to say that
as far as we've looked,
[3:02] we've kept finding twin primes.
[3:04] But of course, that is not how you solve
[3:06] the twin prime conjecture,
[3:07] because you cannot physically check
[3:09] all the numbers out to infinity.
[3:11] So, we need another way.
[3:13] And around 100 years ago,
[3:15] it felt like mathematicians
were getting close
[3:17] with a more sophisticated method.
[3:20] In 1923, English mathematicians,
[3:22] Hardy and Littlewood figured
out how you can estimate
[3:25] how many twin primes there should be.
[3:27] To do it, they started with one
of the crowning achievements
[3:29] of number theory, the
prime number theorem,
[3:32] which tells you that the odds
[3:33] of a large number near N being prime
[3:35] are roughly one over the
natural logarithm of N.
[3:39] So, let's do a quick example
to see how they use this.
[3:43] Say you wanna find the
odds that 137,037 is prime.
[3:47] Well, you just plug that in
[3:49] and find that it has about
an eight and a half percent
[3:51] chance of being prime.
[3:53] Of course, you could use the same trick
[3:54] to find the odds that 137,039 is prime,
[3:58] which is also around 8.5%.
[4:02] So, what are the odds that
both of them are prime?
[4:05] Well, you just multiply
those odds together,
[4:07] that gives you around a 0.7% chance.
[4:10] Now we are simplifying a little here
[4:12] because we're assuming that
primes are independent,
[4:15] which they're not.
[4:16] But putting that to one side,
[4:18] the odds of both a large number N
[4:20] and N plus two being prime,
[4:22] is one over ln(n) times
one over ln(n) plus two,
[4:26] for large N the plus two is insignificant.
[4:29] So, we get the odds of any pair of numbers
[4:32] of size N being a twin prime
[4:34] are roughly one over ln(n) squared.
[4:38] But of course, the
problem with this approach
[4:40] is that the odds of a pair
being a twin prime decrease
[4:43] as you go to larger numbers.
[4:44] So, to count all twin primes up to N,
[4:47] we add up the odds at
every single position.
[4:49] In other words, we
integrate this expression
[4:52] from the first prime
two up to that number.
[4:55] Now Hardy and Littlewood
refine this argument further
[4:58] to account for the fact that primes
[4:59] aren't really independent.
[5:01] But all that added is just
this correction factor upfront.
[5:04] So, the final expression looks like this.
[5:07] (bright music)
[5:08] Now let's plot how many twin primes,
[5:10] this predicts up to 1 trillion.
[5:12] We see that it keeps growing
[5:13] and if we fill in the actual count,
[5:16] you'll notice that it is extremely close
[5:20] to the point where by 1 trillion,
[5:21] our estimate is only off by 0.001%.
[5:26] (bright music)
[5:29] - The only issue though, is
that this is just an estimate
[5:33] or what mathematicians call a heuristic.
[5:36] It can tell you roughly how
many twin primes to expect,
[5:39] but it can't guarantee that they never stop.
[5:42] As Terry Tao puts it, for all we know,
[5:45] there could be this fast conspiracy
[5:47] that every time a number
N decides to be prime,
[5:50] it has some secret agreement
[5:52] with its neighbor N plus two saying
[5:54] you're not allowed to be prime anymore.
[5:57] So, what we need is a rigorous
airtight mathematical proof,
[6:02] but finding such a proof
[6:03] it turns out, is extremely difficult.
[6:07] One of the first mathematicians
[6:08] to really try was a 29-year-old
Norwegian Viggo Brun.
[6:12] (bright music)
[6:14] Brun was working during the early years
[6:16] of the first World War,
[6:17] and with travel disrupted
[6:19] and Europe's mathematical
centers largely out of reach,
[6:22] much of his work happened in
quiet isolation back in Norway.
[6:27] His goal was simple to try
and prove that the number
[6:30] or count of twin primes always increases.
[6:33] And to do this, he took a 2000
year old prime counting tool
[6:37] and tried to adapt it for twin primes.
[6:39] Now the normal version
of this tool is called
[6:41] the sieve of Eratosthenes, and
it works something like this.
[6:46] (bright music)
[6:47] Say we want to find the primes up to 100.
[6:50] First we remove one
because that's not prime.
[6:53] Then we circle the first
prime, that is two,
[6:55] and we remove all of its multiples.
[6:58] We do the same for three.
[6:59] Circle it and remove its multiples,
[7:02] (bright music)
[7:03] and then repeat that for five and seven.
[7:06] And with these four very quick operations,
[7:09] every single number that's
left on your 10 by 10 grid
[7:12] is a prime number.
[7:15] Now we stop the sieve at seven,
[7:17] and we could keep going sieve by 11 or 30,
[7:21] but it turns out that we don't have to.
[7:23] So, why is that?
[7:25] Well, the next prime is 11.
[7:27] So, let's check the numbers below 100
[7:29] that are divisible by 11.
[7:31] 22 is two times 11,
[7:33] but since it's divisible by
two, we already caught it.
[7:36] We also have 33, but that
is divisible by three.
[7:39] And a similar logic holds for 55 and 77.
[7:43] But what about 11 times 11?
[7:46] Well notice that is 121,
which is not in our range.
[7:51] So, every multiple of 11 below 100
[7:54] was already caught by a smaller prime.
[7:57] So, sieving by 11 would
cross out nothing new.
[8:00] - So, Eratosthenes,
[8:02] has this idea that you
only need to sieve out
[8:05] the primes that are at most square root,
[8:06] the size of the number.
[8:08] - But there's also a second
wave to visualize this sieve.
[8:11] Take the number line
[8:13] and let each prime send out a
wave with the wavelength equal
[8:16] to the size of that number.
[8:18] So, the wave from two lands
on every second number
[8:21] and the wave from three
on every third and so on.
[8:24] (bright music)
[8:26] Now, notice that wherever a wave lands,
[8:29] that number has to be composite.
[8:32] But the numbers that escape
every preceding wave,
[8:34] well those have to be prime.
[8:37] And so you end up with this
beautiful visualization
[8:41] of the sieve.
[8:42] (bright music)
[8:43] And both of these methods
make it incredibly easy
[8:46] to identify the primes.
[8:49] - But Brun didn't care about
which primes were left,
[8:52] he just needed to know how many were left.
[8:55] And to do this, he counted all the numbers
[8:58] that were crossed out and then subtracted
[9:00] this from the total,
[9:01] since whatever is not
crossed out must be prime.
[9:05] So, let's use this to find the
number of primes below 100.
[9:09] We can keep a running count.
[9:10] So, we'll start with 100.
[9:12] We're gonna throw out
all the multiples of two.
[9:14] Well, that's 50 numbers left, okay?
[9:16] Then we're gonna throw
away multiples of three.
[9:18] Well, how many multiples of three
[9:19] are there less than 100.
[9:20] 100 divided by three
is not a whole number.
[9:22] It's like 33 and a third, but
we don't throw out fractions.
[9:26] So, okay, forget that
stupid little fraction.
[9:28] We do the same for the
multiples of five and seven,
[9:30] but now the count has gone negative.
[9:33] Well, it turns out that some
numbers got crossed out twice,
[9:36] like six for example, it's
a multiple of two and three.
[9:39] So, we crossed it out once when
we removed multiples of two
[9:42] and again when we removed
multiples of three.
[9:44] So, we have to add it back once.
[9:46] The same thing happens
for multiples of 10,
[9:48] which is two times five,
[9:50] 14 which is two times seven,
[9:51] 15, which is three times five and so on.
[9:54] So, we add these overlaps back
[9:56] and now the count rises to 28,
[9:59] but it's still not quite right.
[10:01] - Now we've triple added
back in the multiples
[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
times five times seven.
[10:15] There's a fourth triple two,
three times five times seven,
[10:17] which is 105, but that's
already bigger than 100.
[10:21] So, it contributes nothing here.
[10:23] But we'll write it in to
keep the pattern complete.
[10:26] After the triples, we
add back the quadruple,
[10:29] two times three times five
times seven, which is 210.
[10:32] That's also bigger than 100,
[10:34] so it contributes zero as well.
[10:36] And finally, we add back the
four sieving primes themselves
[10:40] and subtract one because it
was included in the count.
[10:43] So, the count lands at 25 primes.
[10:47] This process of alternating
[10:48] between subtracting and
adding numbers back in
[10:51] is called inclusion exclusion.
[10:53] And it makes it super easy to find out
[10:54] how many primes there are
up to some given number.
[10:58] Now this equation still looks
a little bit like a mess,
[11:01] but actually we can
rearrange it into this form.
[11:05] Now notice that every prime factor
[11:07] we added just adds an
additional term in this series.
[11:10] We've got the two that goes over here,
[11:13] the three goes over here,
five goes over here,
[11:16] and seven goes over here.
[11:18] And if we were trying to find
primes for some higher number
[11:21] and we wanted to add 11 as a prime factor,
[11:23] it would just add one additional
one minus one over 11 term.
[11:27] Now in general,
[11:28] if we keep the sieve
going up to some prime P,
[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
the count of primes.
[11:40] What Brun really wanted was
the count of twin primes.
[11:43] And so this is the part
where he adapted his sieve,
[11:47] to find a case where both
N and N plus two are prime.
[11:51] For example, when you're sieving by five,
[11:53] you remove the numbers where
N is divisible by five,
[11:56] just as before.
[11:58] But in addition, you need
to remove the numbers
[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
two is 25, 30, 35 and so on.
[12:13] So, this means that while
in the ordinary sieve
[12:16] a prime P removes about
one out of every P numbers,
[12:19] the twin prime sieve removes two.
[12:22] And this changes the count to this,
[12:25] where the numerator for
all the terms after two,
[12:28] ends up getting a two.
[12:31] Now, if we plot the count of
twin primes that this predicts,
[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
by three is 33, and a third,
[12:51] but forget about that little error,
[12:52] those little errors,
[12:54] it's very difficult to
control those errors
[12:56] 'cause there are so many
of those little errors.
[12:58] Each one of them is at most
one it's a rounding error.
[13:00] - Take the traditional
sieve of Eratosthenes again,
[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,
two, three, and five,
[13:15] there are already seven
separate rounding errors,
[13:18] each for one term in
the inclusion exclusion.
[13:22] And so you might start to
see the general pattern here.
[13:25] One prime gives two to the
power one minus one terms,
[13:28] two gives two to the
power two minus one terms,
[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
power K minus one error terms,
[13:42] but we can drop the one
[13:43] because it doesn't really
make much of a difference.
[13:47] So, that means that if you're
sieving for normal primes,
[13:49] the error grows roughly
like two to the power K.
[13:53] But if you're saving for twin
primes, it gets even worse.
[13:56] And now the error grows roughly
like four to the power K.
[13:59] And so this causes a lot of
trouble very, very quick.
[14:03] (bright music)
[14:06] And then here for the twin
primes, it's even worse.
[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
terms start dominating,
[14:25] you get into trouble.
[14:26] So, the issue he was facing
is he had this main term
[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
theory in particular
[14:40] is the fight between the main term
[14:41] that you think is the truth,
if it actually is the truth,
[14:44] and getting the error
term to be provably lower order.
[14:47] - So, for twin primes,
[14:49] the main term grows roughly like this,
[14:51] something like N over the
natural logarithm of N squared.
[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
be anywhere in this range,
[15:08] and as you can see, a big
swath of it is negative.
[15:11] So, to prove the
conjecture, what we must do
[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
terms start to accumulate
[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
way up to square root X,
[15:37] but to up to something less than that,
[15:40] and you are very careful about
your inclusion exclusion,
[15:44] you can actually take
care of those error terms.
[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
over the error term.
[15:56] But this chooses a trade off.
[15:58] Say you wanted to find
all the prime numbers
[16:00] up to 10 billion,
[16:02] then using the square root method,
[16:03] you would need to sieve up to
the square root of 10 billion,
[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
sieve up to 10 billion
[16:16] to the power one over
10, which is just 10.
[16:20] But there's a catch.
[16:21] Because Brun, didn't
sieve by all the primes,
[16:24] there were some survivors
that weren't prime at all,
[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
most nine prime factors.
[16:42] - Brun's techniques were improved
and improved and improved,
[16:45] and we went from nine prime
factors to seven prime factors
[16:49] to three prime factors.
[16:50] Until in 1973,
[16:52] a Chinese mathematician
Chen Jingrun comes along
[16:56] and proved that there are
infinitely many prime P,
[16:58] where P plus two has at
most two prime factors.
[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
like as close as we can get
[17:12] and no further.
[17:14] - Right.
[17:14] So, that's one approximation,
[17:17] one mechanism for
approximating twin primes.
[17:19] There's another.
[17:20] So, the other mechanism
is what's the smallest gap
[17:24] between two consecutive primes?
[17:25] So, instead of saying they differ by two
[17:27] and one of them's prime
and the other one, well,
[17:30] we're gonna try to get,
you know, reduce the number
[17:32] of prime factors that it has.
[17:33] Instead we're gonna say
both numbers must be prime,
[17:36] but let's reduce the
distance between them.
[17:39] - Now remember, on average
consecutive primes sit
[17:42] about the natural logarithm of N apart.
[17:45] And so the question became,
can we at least prove
[17:48] that primes come closer
than that average gap?
[17:52] For decades, mathematicians
kept chipping away
[17:55] at this problem.
[17:56] By 1988, the gap had
dropped all the way down
[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,
Pintz and Yildirim,
[18:14] proved a result that shocked
the mathematical community.
[18:17] - And they announced the
proof of a spectacular result,
[18:20] 0% bounded gaps of 0% of the average.
[18:24] - Wait, what?
[18:26] - They proved that you
can make the fraction
[18:27] as small as you want.
[18:29] This means the gap could
be one 10th or 100th,
[18:33] or even 1000000th of the average gap.
[18:35] It could be as arbitrarily
small as you like,
[18:38] and infinitely often
primes get that close.
[18:42] - When I was young, we didn't
know there were gaps of size,
[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
close to something even bigger,
[18:57] an absolute bounded
gap between two primes.
[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
a concrete bounded gap?
[19:11] - Well, because we think that the bound
[19:13] between two consecutive primes
infinitely often is two,
[19:15] and right now we're showing
that it's log X, and X grows.
[19:18] - If they could get to bounded gaps,
[19:21] then they had another method
to attack the conjecture.
[19:25] The only problem
[19:26] was that it seemed like
their tool ran into a wall.
[19:30] And so in 2005,
[19:32] the American Institute of
Mathematics convened a meeting,
[19:35] gathering all the world's
leading experts on this problem.
[19:39] GPY, Andrew Granville,
Kannan Soundararajan,
[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
to go to this meeting.
[19:56] You're surrounded by
the world's top experts
[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
how it's not possible
[20:07] to do this thing.
[20:08] And so as far as I was
concerned, that was that, and I,
[20:11] you know, this wasn't
gonna be my thesis problem,
[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
who was not at this meeting,
[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
recommendation letters
[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
odd jobs for seven years,
[20:41] including at Subway,
where he kept the books
[20:44] and sorted receipts.
[20:47] Yet, while doing all of
that in his spare time,
[20:50] he would drive down to the local library
[20:52] to read books and
journals on number theory.
[20:56] (bright music)
[20:57] Then in 1999, he got a lucky break.
[21:01] One of his friends helped
him get a job as a lecturer
[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
exactly it is that GPY had built.
[21:35] See, they wanted to find two
primes within a bounded gap.
[21:38] And to do this, they
imagined a stencil of sorts
[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
has a diameter of six
[21:49] and holes at spot zero, two and six.
[21:53] Next we place the stencil
anywhere on the number line,
[21:56] say starting at 12.
[21:59] And we ask a simple question,
how many primes do we see?
[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
over one spot and repeat,
[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
stencil across the number line
[22:29] and it keeps catching two primes,
[22:31] then you have proven
[22:33] that infinitely often
pairs of primes exist
[22:36] within a bounded gap.
[22:37] Six in this case, since that's
the diameter of our stencil.
[22:42] But there are two problems
with this approach.
[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
course you can't keep sliding
[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
where all the primes are,
[23:04] we do know how they behave on average.
[23:07] For example, say we take some
stretch of the number line
[23:10] from some very large number X to two X,
[23:14] then we can't predict
exactly which numbers
[23:16] in this stretch will be prime.
[23:18] But what we can do is put this
into the prime number theorem
[23:21] and estimate how many
primes to expect on average.
[23:25] (bright music)
[23:27] And GPY realized they
could do a similar thing
[23:30] for their stencil method
[23:32] to build an averaging machine of sorts.
[23:35] All you do is you set
up the stencil you want,
[23:37] input your range, and out comes
the average number of primes
[23:40] it caught per position.
[23:42] So, let's do a quick
example without the machine
[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
use a much larger range,
[23:55] but for simplicity, let's stick to this.
[23:58] To find the average, you
just place the stencil
[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
increase the total by one.
[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
across 21 starting positions.
[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
checked every spot by hand,
[24:55] we could see that the stencil
caught several positions
[24:58] with two primes in them.
[25:00] But the averaging machine
doesn't tell us this.
[25:03] All that spits out is an average,
[25:05] and that average alone
[25:07] can't guarantee that it
caught two primes at once,
[25:10] because you could just
as well imagine spreading
[25:12] all of these 14 primes over
14 individual positions.
[25:17] And the same would be true
if the average was 0.8, 0.9,
[25:21] or even one.
[25:23] I know it's unlikely,
[25:24] but it could be that every
position only caught one prime
[25:28] to bring the average to one.
[25:31] But what if the average
was larger than one?
[25:34] What if we found 22 primes in total?
[25:37] Well, even if every position
had caught one prime,
[25:41] you would still have one prime left over
[25:43] and that needs to go in one
of those other positions.
[25:47] So, in other words, if
the average is above one,
[25:50] then that guarantees
[25:51] that at least one position
caught two primes.
[25:55] And so that's the game.
[25:56] Get the average number
of primes above one,
[25:59] but that still leaves one question.
[26:02] Even if you could do that, then
how does that help you prove
[26:06] that you keep getting bounded gaps
[26:08] all the way up to infinity?
[26:10] Well, remember you just proved
that for any large number X,
[26:15] so there's nothing stopping
you from replacing that X
[26:18] with two X, and that two X with four X,
[26:21] and then you could replace
the two X with four X
[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
the way out to infinity,
[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
tweak to their machine.
[26:59] They realize that some starting positions
[27:01] are more likely to catch
primes than others.
[27:04] For example, if the stencil
only lands on even numbers,
[27:07] it will always give zero
primes unless it includes two.
[27:11] So, those positions
[27:12] should not be weighted
equally to the others.
[27:15] Similarly, if you can divide
one or more of the numbers
[27:17] in the stencil by three, five, seven,
[27:19] or other small prime factors,
[27:21] it's also more unlikely you
have primes in your stencil.
[27:24] So, they should also be weighted less.
[27:27] Now, GPY did a few simple checks
like this for each position
[27:30] to assign it a weight,
and estimate of how likely
[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
weighted average shot up.
[27:43] With their updated machine
[27:45] they proved that if you
pick any large number X,
[27:48] then in the range X to two X,
[27:50] you could get the weighted
average arbitrarily close to one,
[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
weigh different positions,
[28:04] they needed to know how
primes are distributed
[28:07] in something known as an
arithmetic progression.
[28:09] These are basically just series of numbers
[28:12] where they're all the
same step size apart.
[28:15] So 3 7 11 15
[28:17] that's an arithmetic progression
with a step size of four.
[28:21] Now, GPY needed to know
how primes were distributed
[28:24] over many such arithmetic progressions
[28:26] with all kinds of different step sizes.
[28:29] Fortunately, they knew
of theorem from the 1960s
[28:32] they could use to do exactly this.
[28:34] - It's the substitute for the
Riemann hypothesis on average,
[28:37] and it turns out that that's
what's actually necessary
[28:40] in a lot of these arithmetic applications.
[28:42] - But that theorem
doesn't work everywhere.
[28:45] It only works for step sizes
up to the square root of X
[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
a reliable count of primes
[29:00] in those progressions.
[29:02] And mathematicians believed
[29:03] that the theorem worked
beyond X to the half,
[29:06] but this was never proven,
[29:08] and that is where the trouble lies.
[29:11] When GPY ran their
calculation with this ceiling,
[29:14] they found the maximum
weighted average they could get
[29:17] was two times theta.
[29:19] In other words, they
got really close to one,
[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
0% of the average gap,
[29:34] they can get bounded gaps.
[29:36] The primes would be some
fixed distance apart.
[29:38] - That's what the 2005
meeting was all about.
[29:41] They kept trying to push
past that one half barrier,
[29:45] but they ended up concluding
that it was impossible.
[29:49] From 2010 on, Zhang spent
two years hacking away
[29:52] at it as well, internalizing
the GPY argument,
[29:56] working day and night just to
try and find a way through.
[30:00] (bright music)
[30:02] But by the summer of 2012,
[30:04] Zhang was exhausted, and
had nothing to show for it.
[30:08] Hoping to clear his head, he
visited a friend in Colorado,
[30:12] where one evening while they were waiting
[30:14] to leave for concert,
[30:15] Zhang stepped outside
alone into the backyard,
[30:19] looking out for the deer that
usually roamed the property,
[30:23] but no deer came.
[30:26] So, he was just walking and thinking
[30:28] when all of a sudden
the answer came to him.
[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
special class of step sizes,
[30:49] one's built only from small prime factors,
[30:52] and he realized he could
reorganize the error terms
[30:55] into a form where most
of them get canceled out.
[30:58] This allowed him to push
past the half barrier
[31:00] by a tiny fraction, just one over 584.
[31:06] Zhang spent the next year
finalizing his proof,
[31:09] and then on the 17th of April, 2013,
[31:11] he sent off that curious email
to the Annals of Mathematics.
[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
this isn't gonna work out,
[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
know, sorry, it didn't work out.
[31:33] Here's where the the gap is.
[31:35] And they start reading it,
[31:37] and then they flip back,
because they're experts.
[31:39] So, they can just flip through
like, yeah, this'll work,
[31:41] this will work, but wait a second,
[31:42] you're gonna get stuck here.
[31:43] And they flip five pages
and they're like, oh,
[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
make it work over there.
[31:59] And then he goes over there,
no, he cut it just right.
[32:01] It fits in that corner.
[32:02] Wait a second.
[32:03] And then they flip, you
know, five more pages
[32:05] and by the end of the week,
[32:06] they've reconstructed the
proof and everything's right.
[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
slots always catch primes,
[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
unknown in the field.
[32:31] I actually thought
[32:32] when I started reading
Yitang Zhang's paper
[32:33] and started realizing
it was probably correct,
[32:36] I thought it was probably
one of the people
[32:37] I knew under a pseudonym,
[32:39] trying to avoid being
embarrassed by being wrong.
[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
really interesting aspect
[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,
people took it seriously.
[33:08] They looked at it.
[33:09] They didn't think that
he had a true proof.
[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
hero as he should have been.
[33:17] To me, it shows that
mathematics culture works.
[33:19] We're doing the right thing.
[33:21] I think the cover of
like Scientific American
[33:24] or whatever in 2013 is
the story Yitang Zhang
[33:27] and how he got this bounded gap
[33:28] between primes toiling
in complete isolation.
[33:31] And maybe it's because he was in isolation
[33:33] that he didn't have the group
think that the rest of us did.
[33:35] To know that this was not,
[33:37] we were all told it's
an impossible problem.
[33:39] - After realizing a bounded gap
was not an impossible dream,
[33:44] people reworked Zhang's
method to optimize it.
[33:47] Terence Tao spearheaded an
online group called Polymath,
[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
being at a conference
[33:59] at the time, as I remember,
sort of day by day,
[34:02] everyone was refreshing
their screens to see
[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
postdoc named James Maynard,
[34:16] who just got his PhD at
Oxford with Roger Heath-Brown
[34:19] and other one of these analytic
number theory world experts.
[34:22] And he's working with Andrew
Granville, in Montreal.
[34:26] And he has a completely orthogonal
approach to this problem
[34:31] that he'd been making
[34:32] some very incremental
progress on independently.
[34:34] - When Maynard started, his
advisor explicitly told him,
[34:38] "I hope you won't work on
this problem full time,
[34:41] because I'm really pretty
confident you're going to fail."
[34:45] But Maynard ignored the warning,
[34:47] and came up with a different
way to attack the problem,
[34:51] and within just a few
months, he got it to work.
[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
in a bounded window.
[35:03] The bound has to change
depending on how many primes
[35:05] you wanna put in that window,
[35:06] and that his number is better.
[35:08] And the method has nothing to
do with the exponent one half.
[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
fundamental limit at all.
[35:22] See where GPY average was
stuck at two times theta,
[35:26] Maynard's average grew
roughly like theta over two
[35:29] times the natural logarithm of K,
[35:31] where K is the number
of slots in the stencil.
[35:34] So, all Maynard needed was
enough slots and it worked.
[35:39] - You need any number greater than zero.
[35:41] When you go up to one half
because you have one half,
[35:43] you'll get better numerics.
[35:45] But the bounded gaps you
could do, just going up to,
[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
with Andrew Granville,
[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
doing exactly the same thing.
[36:09] We gotta get these two guys
to talk to one another.
[36:11] Terry's a Fields medalist,
[36:13] he's like a super
heavyweight by this point,
[36:16] whereas James is this,
you know, fresh PhD,
[36:21] and Terry says, you take
it, I don't need this.
[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
was just a proof of concept,
[36:34] and the same methods would
give something smaller,
[36:37] but there were extra ideas
that you could maybe use
[36:40] to squeeze everything out.
[36:42] And so the current world record
[36:43] is that there's infinitely
many pairs of primes
[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
awarded the Fields Medal,
[36:56] Mathematics, highest honor
for his work on prime gaps.
[37:00] - Yeah, I think that's the basic outline
[37:03] is the kind of two
directions of approximation
[37:06] to twin primes Chen's direction,
[37:08] Brun, Chen, whatever,
versus Zhang and Maynard,
[37:12] and this story that the
Zhang wasn't at this meeting,
[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
Landy broke it as well.
[37:32] And by the end of 1956,
10 people had broken it,
[37:37] all from knowing that it was possible.
[37:41] So, what about pushing
that gap down even lower?
[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
Elliot-Halberstam conjecture,
[37:56] which assumes primes
[37:57] are spread evenly across
arithmetic progressions,
[38:00] or that the level of distribution
[38:02] can be taken as large as one.
[38:04] And in 2013, Maynard
showed that if you assume
[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
proved that if you assume
[38:15] an even stronger version
of this conjecture,
[38:17] the gap drops all the way down to six.
[38:21] But without assuming
anything, 246 still stands.
[38:27] Let me ask you, do you
think we're gonna solve
[38:30] the twin Prime conjecture?
[38:31] - I am totally convinced that
humanity will eventually solve
[38:36] the twin prime conjecture.
[38:37] Often when I'm asked about some
of these big famous problems
[38:40] like Twin Primes or Riemann
or something like that,
[38:45] one way of kind of avoiding the question
[38:48] is that if you imagine you
are randomly distributed in time,
[38:52] you'd expect to be sort of
typically roughly halfway
[38:57] through when the
conjectures been open for.
[38:59] So, an actual guess, if you
have no other information,
[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
work for twin primes,
[39:09] 'cause we don't know whether it's-
[39:11] - Right.
[39:12] (all laughing)
[39:13] - 125 years old or whether
it's like 2000 years old.
[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
certainty is for the best.
[39:30] Because if we knew for sure
that this was impossible,
[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
not to know everything.
[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
would help everything click.
[40:00] You know, the fastest
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[40:02] is to have someone next to you
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