---
title: 'The Infinite Twin Prime Problem'
source: 'https://youtube.com/watch?v=8HBDE-msUjw'
video_id: '8HBDE-msUjw'
date: 2026-06-14
duration_sec: 2490
---

# The Infinite Twin Prime Problem

> Source: [The Infinite Twin Prime Problem](https://youtube.com/watch?v=8HBDE-msUjw)

## Summary

The video explores the history and mathematics behind the twin prime conjecture, which posits that there are infinitely many pairs of primes separated by two. It covers key attempts to prove the conjecture, from Viggo Brun's sieve to Yitang Zhang's breakthrough showing a bounded gap of 70 million, and James Maynard's subsequent improvements. The narrative highlights the interplay between heuristic reasoning, rigorous proof, and the role of unknown mathematicians.

### Key Points

- **The Email That Shook Mathematics** [00:00] — In 2013, the Annals of Mathematics received a proof from an unknown mathematician, Yitang Zhang, claiming a breakthrough on the twin prime conjecture.
- **Twin Prime Conjecture Explained** [01:06] — Twin primes are pairs of primes separated by one number (e.g., 11 and 13). The conjecture states there are infinitely many such pairs.
- **Hardy-Littlewood Heuristic** [03:13] — In 1923, Hardy and Littlewood estimated the density of twin primes using the prime number theorem, predicting the count grows like N/(log N)^2.
- **Viggo Brun's Sieve** [06:07] — Brun adapted the sieve of Eratosthenes to twin primes, but error terms grew too fast. He proved there are infinitely many pairs with at most nine prime factors.
- **Chen's Theorem** [16:52] — In 1973, Chen Jingrun proved there are infinitely many primes p such that p+2 has at most two prime factors, the closest approach to the conjecture.
- **GPY Breakthrough and Stencil Method** [19:01] — Goldston, Pintz, and Yildirim developed a method using a stencil to find primes within bounded gaps, but their weighted average could not exceed one due to the level of distribution limit.
- **Zhang's Insight** [30:41] — Zhang focused on step sizes with only small prime factors, reorganizing error terms to cancel out, pushing past the half barrier by 1/584.
- **Zhang's Result: Gap of 70 Million** [32:16] — Zhang proved a bounded gap of 70 million, showing infinitely many prime pairs within that distance.
- **Polymath and Maynard's Improvements** [33:47] — Terence Tao's Polymath project reduced the gap to 4,680. James Maynard independently developed a method that lowered it to 600 and later to 246.
- **Current Record and Conditional Results** [36:49] — The unconditional record is 246. Assuming the Elliott-Halberstam conjecture, the gap can be reduced to 12 or even 6.

### Conclusion

The twin prime conjecture remains unsolved, but progress has been made through sieves, the GPY method, and Zhang's breakthrough. The current unconditional bound is 246, and further improvements may require new ideas.

## Transcript

- [Derek] On the morning
of the 17th of April, 2013,
the journal Annals of Mathematics
received a curious email.
It claimed to contain a 50
page proof relating to one
of the oldest unsolved
problems in mathematics.
A problem that great
mathematician Edmund Landau
called unattackable.
But this proof didn't come
from a famous professor,
it came from an unknown.
(bright music)
Someone who had once spent years working
at a subway restaurant.
- So, they're like, okay,
surely this isn't gonna
work out, but whatever.
You know, we'll send it to a referee.
- [Derek] They expected to
find a mistake in an afternoon,
but they didn't.
So, they went through it again,
closely studying the fragile parts
where a proof like this
normally falls apart,
but still nothing.
Soon they realized they were
witnessing a breakthrough.
- Oh, damn, he did it.
- So, how did he do It?
What did the experts miss
and what was the problem
he was working on?
He was working on a new way
to attack one of the
hardest unsolved problems
in number theory,
the twin prime conjecture.
The twin primes
are prime numbers separated
by just one number,
like 11 and 13 or 17 and 19.
As you go up, the number
line, primes become rarer
and twin primes become rarer still.
But the twin prime conjecture claims
that there are infinitely many
of them you never run out.
But is it true?
Well, one way to approach this problem
is to look at the gaps
between consecutive primes
as you go up the number line.
(bright music)
Now, at first they seem chaotic,
but if you average them
out, a clear trend emerges.
The average gap between
two primes grows roughly
as the natural logarithm of the number N.
So, for example, the average gap
between primes around
100, is approximately 4.6.
The average gap between
primes around 1000 is 6.9.
Logarithms grow very slowly,
but they do keep growing forever.
So, as N approaches infinity,
the average gap between
primes also goes to infinity,
which is not particularly encouraging
if you expect to always
be able to find primes
that are just two apart.
But if you start checking large numbers,
then after a million you
quickly find the twin primes,
1,000,037 and 1,000,039.
Past a billion, there's
1,000,000,007 and 1,000,000,009.
In fact, we have found a
twin prime that's as large
as 2,996,863,034,895
times two to the power of 1,290,000
plus or minus one.
That is a pair of numbers
each 388,342 digits long.
If you wanted to print those
in a book for some reason,
you would need around
260 pages per number.
All of this is to say that
as far as we've looked,
we've kept finding twin primes.
But of course, that is not how you solve
the twin prime conjecture,
because you cannot physically check
all the numbers out to infinity.
So, we need another way.
And around 100 years ago,
it felt like mathematicians
were getting close
with a more sophisticated method.
In 1923, English mathematicians,
Hardy and Littlewood figured
out how you can estimate
how many twin primes there should be.
To do it, they started with one
of the crowning achievements
of number theory, the
prime number theorem,
which tells you that the odds
of a large number near N being prime
are roughly one over the
natural logarithm of N.
So, let's do a quick example
to see how they use this.
Say you wanna find the
odds that 137,037 is prime.
Well, you just plug that in
and find that it has about
an eight and a half percent
chance of being prime.
Of course, you could use the same trick
to find the odds that 137,039 is prime,
which is also around 8.5%.
So, what are the odds that
both of them are prime?
Well, you just multiply
those odds together,
that gives you around a 0.7% chance.
Now we are simplifying a little here
because we're assuming that
primes are independent,
which they're not.
But putting that to one side,
the odds of both a large number N
and N plus two being prime,
is one over ln(n) times
one over ln(n) plus two,
for large N the plus two is insignificant.
So, we get the odds of any pair of numbers
of size N being a twin prime
are roughly one over ln(n) squared.
But of course, the
problem with this approach
is that the odds of a pair
being a twin prime decrease
as you go to larger numbers.
So, to count all twin primes up to N,
we add up the odds at
every single position.
In other words, we
integrate this expression
from the first prime
two up to that number.
Now Hardy and Littlewood
refine this argument further
to account for the fact that primes
aren't really independent.
But all that added is just
this correction factor upfront.
So, the final expression looks like this.
(bright music)
Now let's plot how many twin primes,
this predicts up to 1 trillion.
We see that it keeps growing
and if we fill in the actual count,
you'll notice that it is extremely close
to the point where by 1 trillion,
our estimate is only off by 0.001%.
(bright music)
- The only issue though, is
that this is just an estimate
or what mathematicians call a heuristic.
It can tell you roughly how
many twin primes to expect,
but it can't guarantee that they never stop.
As Terry Tao puts it, for all we know,
there could be this fast conspiracy
that every time a number
N decides to be prime,
it has some secret agreement
with its neighbor N plus two saying
you're not allowed to be prime anymore.
So, what we need is a rigorous
airtight mathematical proof,
but finding such a proof
it turns out, is extremely difficult.
One of the first mathematicians
to really try was a 29-year-old
Norwegian Viggo Brun.
(bright music)
Brun was working during the early years
of the first World War,
and with travel disrupted
and Europe's mathematical
centers largely out of reach,
much of his work happened in
quiet isolation back in Norway.
His goal was simple to try
and prove that the number
or count of twin primes always increases.
And to do this, he took a 2000
year old prime counting tool
and tried to adapt it for twin primes.
Now the normal version
of this tool is called
the sieve of Eratosthenes, and
it works something like this.
(bright music)
Say we want to find the primes up to 100.
First we remove one
because that's not prime.
Then we circle the first
prime, that is two,
and we remove all of its multiples.
We do the same for three.
Circle it and remove its multiples,
(bright music)
and then repeat that for five and seven.
And with these four very quick operations,
every single number that's
left on your 10 by 10 grid
is a prime number.
Now we stop the sieve at seven,
and we could keep going sieve by 11 or 30,
but it turns out that we don't have to.
So, why is that?
Well, the next prime is 11.
So, let's check the numbers below 100
that are divisible by 11.
22 is two times 11,
but since it's divisible by
two, we already caught it.
We also have 33, but that
is divisible by three.
And a similar logic holds for 55 and 77.
But what about 11 times 11?
Well notice that is 121,
which is not in our range.
So, every multiple of 11 below 100
was already caught by a smaller prime.
So, sieving by 11 would
cross out nothing new.
- So, Eratosthenes,
has this idea that you
only need to sieve out
the primes that are at most square root,
the size of the number.
- But there's also a second
wave to visualize this sieve.
Take the number line
and let each prime send out a
wave with the wavelength equal
to the size of that number.
So, the wave from two lands
on every second number
and the wave from three
on every third and so on.
(bright music)
Now, notice that wherever a wave lands,
that number has to be composite.
But the numbers that escape
every preceding wave,
well those have to be prime.
And so you end up with this
beautiful visualization
of the sieve.
(bright music)
And both of these methods
make it incredibly easy
to identify the primes.
- But Brun didn't care about
which primes were left,
he just needed to know how many were left.
And to do this, he counted all the numbers
that were crossed out and then subtracted
this from the total,
since whatever is not
crossed out must be prime.
So, let's use this to find the
number of primes below 100.
We can keep a running count.
So, we'll start with 100.
We're gonna throw out
all the multiples of two.
Well, that's 50 numbers left, okay?
Then we're gonna throw
away multiples of three.
Well, how many multiples of three
are there less than 100.
100 divided by three
is not a whole number.
It's like 33 and a third, but
we don't throw out fractions.
So, okay, forget that
stupid little fraction.
We do the same for the
multiples of five and seven,
but now the count has gone negative.
Well, it turns out that some
numbers got crossed out twice,
like six for example, it's
a multiple of two and three.
So, we crossed it out once when
we removed multiples of two
and again when we removed
multiples of three.
So, we have to add it back once.
The same thing happens
for multiples of 10,
which is two times five,
14 which is two times seven,
15, which is three times five and so on.
So, we add these overlaps back
and now the count rises to 28,
but it's still not quite right.
- Now we've triple added
back in the multiples
of two times three times five,
so that we need to subtract one more time.
- The same thing happens for 42,
which is two times three times seven,
and 70, which is two
times five times seven.
There's a fourth triple two,
three times five times seven,
which is 105, but that's
already bigger than 100.
So, it contributes nothing here.
But we'll write it in to
keep the pattern complete.
After the triples, we
add back the quadruple,
two times three times five
times seven, which is 210.
That's also bigger than 100,
so it contributes zero as well.
And finally, we add back the
four sieving primes themselves
and subtract one because it
was included in the count.
So, the count lands at 25 primes.
This process of alternating
between subtracting and
adding numbers back in
is called inclusion exclusion.
And it makes it super easy to find out
how many primes there are
up to some given number.
Now this equation still looks
a little bit like a mess,
but actually we can
rearrange it into this form.
Now notice that every prime factor
we added just adds an
additional term in this series.
We've got the two that goes over here,
the three goes over here,
five goes over here,
and seven goes over here.
And if we were trying to find
primes for some higher number
and we wanted to add 11 as a prime factor,
it would just add one additional
one minus one over 11 term.
Now in general,
if we keep the sieve
going up to some prime P,
which is less than the square root of N,
then the count becomes approximately this.
- But this is still just
the count of primes.
What Brun really wanted was
the count of twin primes.
And so this is the part
where he adapted his sieve,
to find a case where both
N and N plus two are prime.
For example, when you're sieving by five,
you remove the numbers where
N is divisible by five,
just as before.
But in addition, you need
to remove the numbers
where N plus two is divisible by five.
So, you also remove 23, 28, 33 and so on
because in those cases, N plus
two is 25, 30, 35 and so on.
So, this means that while
in the ordinary sieve
a prime P removes about
one out of every P numbers,
the twin prime sieve removes two.
And this changes the count to this,
where the numerator for
all the terms after two,
ends up getting a two.
Now, if we plot the count of
twin primes that this predicts,
we see that it grows roughly like N
over the natural logarithm of N squared,
just as for our heuristic.
And so it seems like Brun did it,
but unfortunately that is not the case.
- We said that 100 divided
by three is 33, and a third,
but forget about that little error,
those little errors,
it's very difficult to
control those errors
'cause there are so many
of those little errors.
Each one of them is at most
one it's a rounding error.
- Take the traditional
sieve of Eratosthenes again,
when we're sieving with just one prime,
we just have an over two.
So, one rounding error,
with two primes, two and three,
there are three separate rounding errors,
but with three primes,
two, three, and five,
there are already seven
separate rounding errors,
each for one term in
the inclusion exclusion.
And so you might start to
see the general pattern here.
One prime gives two to the
power one minus one terms,
two gives two to the
power two minus one terms,
and three primes gives two
to the power three minus one terms.
In general, if you sieve by K primes,
you get roughly two to the
power K minus one error terms,
but we can drop the one
because it doesn't really
make much of a difference.
So, that means that if you're
sieving for normal primes,
the error grows roughly
like two to the power K.
But if you're saving for twin
primes, it gets even worse.
And now the error grows roughly
like four to the power K.
And so this causes a lot of
trouble very, very quick.
(bright music)
And then here for the twin
primes, it's even worse.
The main term grows a bit more slowly,
but the error term grows much faster.
So, it still starts off slow.
And so you can quickly see
that once these error
terms start dominating,
you get into trouble.
So, the issue he was facing
is he had this main term
and the main term, you know,
it grew in the way he wanted to,
but then it just got overtaken
and dominated by those error terms.
- Exactly.
And that's all of analysis,
in general analytic number
theory in particular
is the fight between the main term
that you think is the truth,
if it actually is the truth,
and getting the error
term to be provably lower order.
- So, for twin primes,
the main term grows roughly like this,
something like N over the
natural logarithm of N squared.
But that's not the true count,
because to get the true count,
we must add and subtract the error terms,
which gives us this upper and lower bound.
And now you can see the problem.
Because the true count can
be anywhere in this range,
and as you can see, a big
swath of it is negative.
So, to prove the
conjecture, what we must do
is we must show that this lower bound
is always positive and growing.
But that is where we run into a problem
with the sieve we've been using.
Because the more primes we sieve by,
the more those error
terms start to accumulate
and we just can show this.
- And so what Brun eventually realized
is that if you weaken the sieve,
if you don't sieve all the
way up to square root X,
but to up to something less than that,
and you are very careful about
your inclusion exclusion,
you can actually take
care of those error terms.
- So, run a sieve by fewer factors,
only up to N to the power one over 10.
And as a result,
he gained enough control
over the error term.
But this chooses a trade off.
Say you wanted to find
all the prime numbers
up to 10 billion,
then using the square root method,
you would need to sieve up to
the square root of 10 billion,
which is 100,000.
But that gives you a lot of error terms.
But with Brun's method,
you would only need to
sieve up to 10 billion
to the power one over
10, which is just 10.
But there's a catch.
Because Brun, didn't
sieve by all the primes,
there were some survivors
that weren't prime at all,
but numbers with many prime factors,
in Brun's case up to nine.
So, what he actually ended up proving
is that there are infinitely many pairs
of numbers two apart,
where each number has at
most nine prime factors.
- Brun's techniques were improved
and improved and improved,
and we went from nine prime
factors to seven prime factors
to three prime factors.
Until in 1973,
a Chinese mathematician
Chen Jingrun comes along
and proved that there are
infinitely many prime P,
where P plus two has at
most two prime factors.
- This is as close as you could get
to proving the twin prime conjecture
without actually getting there.
We're stuck like we're
like as close as we can get
and no further.
- Right.
So, that's one approximation,
one mechanism for
approximating twin primes.
There's another.
So, the other mechanism
is what's the smallest gap
between two consecutive primes?
So, instead of saying they differ by two
and one of them's prime
and the other one, well,
we're gonna try to get,
you know, reduce the number
of prime factors that it has.
Instead we're gonna say
both numbers must be prime,
but let's reduce the
distance between them.
- Now remember, on average
consecutive primes sit
about the natural logarithm of N apart.
And so the question became,
can we at least prove
that primes come closer
than that average gap?
For decades, mathematicians
kept chipping away
at this problem.
By 1988, the gap had
dropped all the way down
to roughly a quarter of the average gap.
This means that say the average gap is 100
at some enormous scale,
then primes must sometimes come
within about 25 of each other.
But then in 2005, Goldston,
Pintz and Yildirim,
proved a result that shocked
the mathematical community.
- And they announced the
proof of a spectacular result,
0% bounded gaps of 0% of the average.
- Wait, what?
- They proved that you
can make the fraction
as small as you want.
This means the gap could
be one 10th or 100th,
or even 1000000th of the average gap.
It could be as arbitrarily
small as you like,
and infinitely often
primes get that close.
- When I was young, we didn't
know there were gaps of size,
less than say one 10th log X
or infinitely many primes pairs
and Goldston and Yildirim,
came up with a method to attack that.
- But the method also came
close to something even bigger,
an absolute bounded
gap between two primes.
(bright music)
- And so that was like a big thing.
- What was the big desire to go from 0%
or arbitrarily small to
a concrete bounded gap?
- Well, because we think that the bound
between two consecutive primes
infinitely often is two,
and right now we're showing
that it's log X, and X grows.
- If they could get to bounded gaps,
then they had another method
to attack the conjecture.
The only problem
was that it seemed like
their tool ran into a wall.
And so in 2005,
the American Institute of
Mathematics convened a meeting,
gathering all the world's
leading experts on this problem.
GPY, Andrew Granville,
Kannan Soundararajan,
all gathered for a week in California
with one explicit goal,
to prove a bounded gap between primes.
- I was a young graduate student,
I was very lucky to get
to go to this meeting.
You're surrounded by
the world's top experts
on this subject.
We spent an entire week.
And the upshot of this week
was basically that it's impossible.
And Soundararajan shows
how it's not possible
to do this thing.
And so as far as I was
concerned, that was that, and I,
you know, this wasn't
gonna be my thesis problem,
I'd have to do something else
and I went off and and did other things.
But there was one person
who was not at this meeting,
Yitang Zhang.
(bright music)
- Zhang grew up in China,
and around the age of 30 moved to the U.S.
to get his PhD in mathematics,
but he never got any
recommendation letters
and so struggled to find a job.
He ended up living in his car for time
and ultimately working
odd jobs for seven years,
including at Subway,
where he kept the books
and sorted receipts.
Yet, while doing all of
that in his spare time,
he would drive down to the local library
to read books and
journals on number theory.
(bright music)
Then in 1999, he got a lucky break.
One of his friends helped
him get a job as a lecturer
at the University of New Hampshire.
So, now he could spend all his time doing
what he loved most, math.
Zhang said that when he was a kid,
he "Imagined there would be a day
that I would solve a major math problem."
And by 2010, he had identified
his major problem to focus on,
bounded gaps between primes.
But to understand what he was working on,
we need to understand what
exactly it is that GPY had built.
See, they wanted to find two
primes within a bounded gap.
And to do this, they
imagined a stencil of sorts
with some holes in it.
So, let's do the same,
and for the sake of this example,
let's say the stencil
has a diameter of six
and holes at spot zero, two and six.
Next we place the stencil
anywhere on the number line,
say starting at 12.
And we ask a simple question,
how many primes do we see?
In this case, we see 12, 14, and 18,
none of which are prime.
So, we write down zero.
Next we shift the stencil
over one spot and repeat,
13, 15, and 19, two of which are prime.
So, we write down two.
And then we keep doing this.
If you slide it a bit further,
we catch two primes again, 23 and 29.
Now, if you keep moving the
stencil across the number line
and it keeps catching two primes,
then you have proven
that infinitely often
pairs of primes exist
within a bounded gap.
Six in this case, since that's
the diameter of our stencil.
But there are two problems
with this approach.
The first is that we don't know exactly
where all the primes are.
And the second is that of
course you can't keep sliding
a stencil down the number line forever
because well, it's infinitely long.
Fortunately, there is an easy way
to get around that first problem.
See, while we don't know exactly
where all the primes are,
we do know how they behave on average.
For example, say we take some
stretch of the number line
from some very large number X to two X,
then we can't predict
exactly which numbers
in this stretch will be prime.
But what we can do is put this
into the prime number theorem
and estimate how many
primes to expect on average.
(bright music)
And GPY realized they
could do a similar thing
for their stencil method
to build an averaging machine of sorts.
All you do is you set
up the stencil you want,
input your range, and out comes
the average number of primes
it caught per position.
So, let's do a quick
example without the machine
to see how this helps.
Let's use the same stencil from before
and use the range 20 to 40.
Now, in reality, you would
use a much larger range,
but for simplicity, let's stick to this.
To find the average, you
just place the stencil
at the start of your range,
and you'll look at which numbers you see,
20, 22 and 26.
None of which are prime
and so the total prime count is zero.
Then we shift the stencil over one spot
and see 21, 23 and 27.
One of which is prime, so we
increase the total by one.
Now let's keep shifting the stencil
and keep adding all the primes we see.
No primes, so add zero,
two primes, add two, and so on.
(stencil shuffling)
Now, if you keep shifting the stencil
all the way until the end,
you find a total of 14 primes
across 21 starting positions.
(bright music)
And so the average is 14 over 21,
or about 0.67 primes per location.
Now in this case where we
checked every spot by hand,
we could see that the stencil
caught several positions
with two primes in them.
But the averaging machine
doesn't tell us this.
All that spits out is an average,
and that average alone
can't guarantee that it
caught two primes at once,
because you could just
as well imagine spreading
all of these 14 primes over
14 individual positions.
And the same would be true
if the average was 0.8, 0.9,
or even one.
I know it's unlikely,
but it could be that every
position only caught one prime
to bring the average to one.
But what if the average
was larger than one?
What if we found 22 primes in total?
Well, even if every position
had caught one prime,
you would still have one prime left over
and that needs to go in one
of those other positions.
So, in other words, if
the average is above one,
then that guarantees
that at least one position
caught two primes.
And so that's the game.
Get the average number
of primes above one,
but that still leaves one question.
Even if you could do that, then
how does that help you prove
that you keep getting bounded gaps
all the way up to infinity?
Well, remember you just proved
that for any large number X,
so there's nothing stopping
you from replacing that X
with two X, and that two X with four X,
and then you could replace
the two X with four X
and the four X with eight X, and so,
on all the way up to infinity.
So, if you can prove that the average gap
is above one for any large number X,
then with this clever setup,
it immediately extends all
the way out to infinity,
and you would've proven your bounded gap.
(bright music)
- Unfortunately, running the machine
with just the current inputs,
the stencil visits many positions
where it catches zero primes,
this drags the average down
and means it will never get above one.
So, GPY made one final
tweak to their machine.
They realize that some starting positions
are more likely to catch
primes than others.
For example, if the stencil
only lands on even numbers,
it will always give zero
primes unless it includes two.
So, those positions
should not be weighted
equally to the others.
Similarly, if you can divide
one or more of the numbers
in the stencil by three, five, seven,
or other small prime factors,
it's also more unlikely you
have primes in your stencil.
So, they should also be weighted less.
Now, GPY did a few simple checks
like this for each position
to assign it a weight,
and estimate of how likely
it is that this position contains primes.
This turned their averaging machine
into a weighted averaging machine.
And as a result, that
weighted average shot up.
With their updated machine
they proved that if you
pick any large number X,
then in the range X to two X,
you could get the weighted
average arbitrarily close to one,
but not above one.
They were close, but they ran into a wall.
A wall that came from their weights,
because to figure out how to
weigh different positions,
they needed to know how
primes are distributed
in something known as an
arithmetic progression.
These are basically just series of numbers
where they're all the
same step size apart.
So 3 7 11 15
that's an arithmetic progression
with a step size of four.
Now, GPY needed to know
how primes were distributed
over many such arithmetic progressions
with all kinds of different step sizes.
Fortunately, they knew
of theorem from the 1960s
they could use to do exactly this.
- It's the substitute for the
Riemann hypothesis on average,
and it turns out that that's
what's actually necessary
in a lot of these arithmetic applications.
- But that theorem
doesn't work everywhere.
It only works for step sizes
up to the square root of X
or X to the one half.
The technical term for this one half
is the level of distribution or theta.
It tells you how large the step sizes
can be while still getting
a reliable count of primes
in those progressions.
And mathematicians believed
that the theorem worked
beyond X to the half,
but this was never proven,
and that is where the trouble lies.
When GPY ran their
calculation with this ceiling,
they found the maximum
weighted average they could get
was two times theta.
In other words, they
got really close to one,
but they could never actually cross it.
- They showed in their method
that if you could get beyond a half,
you could do 0.5000001.
Then instead of just getting
0% of the average gap,
they can get bounded gaps.
The primes would be some
fixed distance apart.
- That's what the 2005
meeting was all about.
They kept trying to push
past that one half barrier,
but they ended up concluding
that it was impossible.
From 2010 on, Zhang spent
two years hacking away
at it as well, internalizing
the GPY argument,
working day and night just to
try and find a way through.
(bright music)
But by the summer of 2012,
Zhang was exhausted, and
had nothing to show for it.
Hoping to clear his head, he
visited a friend in Colorado,
where one evening while they were waiting
to leave for concert,
Zhang stepped outside
alone into the backyard,
looking out for the deer that
usually roamed the property,
but no deer came.
So, he was just walking and thinking
when all of a sudden
the answer came to him.
He hadn't brought any notes or paper,
but Zhang believed his idea would work.
(bright music)
- You see, GPY had to count primes
in many different arithmetic progressions
with all sorts of step sizes.
But Zhang focused on a
special class of step sizes,
one's built only from small prime factors,
and he realized he could
reorganize the error terms
into a form where most
of them get canceled out.
This allowed him to push
past the half barrier
by a tiny fraction, just one over 584.
Zhang spent the next year
finalizing his proof,
and then on the 17th of April, 2013,
he sent off that curious email
to the Annals of Mathematics.
- Now, the Annals gets a proof
of the Riemann Hypothesis every other day,
so they're like, okay, surely
this isn't gonna work out,
but whatever.
You know, we'll send it to a referee.
And they're like, well,
let's spend a few hours this afternoon.
We'll find a mistake.
We'll tell the annals, you
know, sorry, it didn't work out.
Here's where the the gap is.
And they start reading it,
and then they flip back,
because they're experts.
So, they can just flip through
like, yeah, this'll work,
this will work, but wait a second,
you're gonna get stuck here.
And they flip five pages
and they're like, oh,
that's interesting.
That's how you're gonna handle that.
Okay, but then you're gonna have another,
it's like trying to lay down a carpet
in a room.
You're like, okay, I know that corner,
that corner's gonna screw you up
even if you managed to
make it work over there.
And then he goes over there,
no, he cut it just right.
It fits in that corner.
Wait a second.
And then they flip, you
know, five more pages
and by the end of the week,
they've reconstructed the
proof and everything's right.
- Wow.
(bright music)
- Zhang's stencil had 3.5 million slots,
spread across a span of 70 million.
And by proving two of those
slots always catch primes,
he proved a bounded gap of 70 million.
When the news broke,
mathematicians were in disbelief.
- It was basically an
unknown in the field.
I actually thought
when I started reading
Yitang Zhang's paper
and started realizing
it was probably correct,
I thought it was probably
one of the people
I knew under a pseudonym,
trying to avoid being
embarrassed by being wrong.
If they were wrong.
- Of course it wasn't, Zhang was real.
And just over a year later
he was awarded the MacArthur Genius Grant.
(bright music)
- To me, this is a
really interesting aspect
of mathematics in particular.
I think it's one of the very few fields
where we have a truly honest approach
to success and what counts as success.
He sent in an argument,
people took it seriously.
They looked at it.
They didn't think that
he had a true proof.
They sat there, they gave him the time
that the argument was due,
and he was immediately made a
hero as he should have been.
To me, it shows that
mathematics culture works.
We're doing the right thing.
I think the cover of
like Scientific American
or whatever in 2013 is
the story Yitang Zhang
and how he got this bounded gap
between primes toiling
in complete isolation.
And maybe it's because he was in isolation
that he didn't have the group
think that the rest of us did.
To know that this was not,
we were all told it's
an impossible problem.
- After realizing a bounded gap
was not an impossible dream,
people reworked Zhang's
method to optimize it.
Terence Tao spearheaded an
online group called Polymath,
and they sharpened the method.
So, every month, week or day,
that upper bound kept coming down.
One of the attendees described
being at a conference
at the time, as I remember,
sort of day by day,
everyone was refreshing
their screens to see
who had the world record now,
and ultimately they got the number
all the way down to 4,680.
- Meanwhile, there's a young
postdoc named James Maynard,
who just got his PhD at
Oxford with Roger Heath-Brown
and other one of these analytic
number theory world experts.
And he's working with Andrew
Granville, in Montreal.
And he has a completely orthogonal
approach to this problem
that he'd been making
some very incremental
progress on independently.
- When Maynard started, his
advisor explicitly told him,
"I hope you won't work on
this problem full time,
because I'm really pretty
confident you're going to fail."
But Maynard ignored the warning,
and came up with a different
way to attack the problem,
and within just a few
months, he got it to work.
Further bring down the gap to 600.
But his method also proved something else.
- He can get three primes
in a bounded window.
The bound has to change
depending on how many primes
you wanna put in that window,
and that his number is better.
And the method has nothing to
do with the exponent one half.
- Wait, what?
- One half was a pure mirage.
It was a red herring.
- That is crazy.
One half was not a
fundamental limit at all.
See where GPY average was
stuck at two times theta,
Maynard's average grew
roughly like theta over two
times the natural logarithm of K,
where K is the number
of slots in the stencil.
So, all Maynard needed was
enough slots and it worked.
- You need any number greater than zero.
When you go up to one half
because you have one half,
you'll get better numerics.
But the bounded gaps you
could do, just going up to,
you know, 0.01, not 0.50101.
Now, curiously, Terry Tao independently
has the same approach.
He tells Ben Green about it,
Ben Green is meeting
with Andrew Granville,
and says, "Hey, Terry's got this new idea
that he thinks is gonna get even farther."
And Granville, says, wait a second.
My postdoc, Maynard, is
doing exactly the same thing.
We gotta get these two guys
to talk to one another.
Terry's a Fields medalist,
he's like a super
heavyweight by this point,
whereas James is this,
you know, fresh PhD,
and Terry says, you take
it, I don't need this.
You know, you this,
it's your idea, you go with it.
- By early 2014, Maynard joined forces
with Tao's Polymath group.
- It was clear that somehow 600
was just a proof of concept,
and the same methods would
give something smaller,
but there were extra ideas
that you could maybe use
to squeeze everything out.
And so the current world record
is that there's infinitely
many pairs of primes
that differ by no more than 246.
(bright music)
- And that for now is where it stops.
In 2022, James Maynard was
awarded the Fields Medal,
Mathematics, highest honor
for his work on prime gaps.
- Yeah, I think that's the basic outline
is the kind of two
directions of approximation
to twin primes Chen's direction,
Brun, Chen, whatever,
versus Zhang and Maynard,
and this story that the
Zhang wasn't at this meeting,
so he didn't know it was impossible.
- It reminded me of the four minute mile
where everyone thought it was impossible.
No one did it.
- And then for the first time ever,
Roger Bannister broke it in 1954,
and after knowing it was possible,
just 46 days later,
another runner called John
Landy broke it as well.
And by the end of 1956,
10 people had broken it,
all from knowing that it was possible.
So, what about pushing
that gap down even lower?
Is that possible?
Well, mathematicians have found ways
to bring it down even more,
but all of these results are conditional.
For example, there is the
Elliot-Halberstam conjecture,
which assumes primes
are spread evenly across
arithmetic progressions,
or that the level of distribution
can be taken as large as one.
And in 2013, Maynard
showed that if you assume
this conjecture is true,
then the gap plummets to just 12.
A year later,
the Polymath Group
proved that if you assume
an even stronger version
of this conjecture,
the gap drops all the way down to six.
But without assuming
anything, 246 still stands.
Let me ask you, do you
think we're gonna solve
the twin Prime conjecture?
- I am totally convinced that
humanity will eventually solve
the twin prime conjecture.
Often when I'm asked about some
of these big famous problems
like Twin Primes or Riemann
or something like that,
one way of kind of avoiding the question
is that if you imagine you
are randomly distributed in time,
you'd expect to be sort of
typically roughly halfway
through when the
conjectures been open for.
So, an actual guess, if you
have no other information,
is to guess that a problem will be open
for as long as it has been open for already.
But obviously this doesn't
work for twin primes,
'cause we don't know whether it's-
- Right.
(all laughing)
- 125 years old or whether
it's like 2000 years old.
So, I think it's a fools game to guess,
but it clearly needs a really big idea,
but maybe it only needs one big idea.
- Although maybe, just maybe,
not knowing this with
certainty is for the best.
Because if we knew for sure
that this was impossible,
then we would've likely missed out
on most of these inventions
and new methods over the past century.
So, sometimes it pays
not to know everything.
- You know, before I started this channel,
I was a teacher at a tutoring company,
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