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00:00In Xiamen, China, in the winter of 1954, air raid sirens are sounded all over the city.
00:08People scramble for cover as artillery shells are being fired in the distance.
00:13The People's Republic of China is bombarding the nearby Kinmen Islands in an attempt to
00:17take over control from the anti-communist Chinese.
00:21But troops on the islands fire back.
00:24Soon, artillery shells are raining down on both sides.
00:28Back in Xiamen, inside one of these shelters is 21-year-old Chen Jingrun.
00:34As others are fidgeting, impatiently waiting for the explosions to stop, Chen is reading
00:39pages from a math book.
00:43Because he's on a mission to solve one of the oldest problems in math.
00:47A child can understand the statement, but the greatest geniuses in mathematical history
00:52have not been able to solve it.
00:54In the year 2000, a publishing house even offered a $1 million prize to anyone who could solve
01:00the problem.
01:01But for nearly 300 years, everyone who has tried has failed.
01:07So the problem is this.
01:09Can every even number greater than 2 be written as the sum of 2 primes?
01:14So that's the problem.
01:16So we can do some examples.
01:18We can say, for example, 6, you know, that's an even integer.
01:22So can we write it as the sum of 2 primes?
01:24So those are numbers that only divide by themselves and by 1.
01:28So for 6, it's just 3 plus 3.
01:31And we can do more and more numbers.
01:33So 10, for example, would be 5 plus 5.
01:37But also 7 plus 3.
01:40There's one I want to do, one extra, which is your favorite number.
01:4442.
01:4542.
01:46What makes it up?
01:4737 and 5.
01:4837.
01:49Yeah.
01:50But there's also a sort of nicer, more methodical way of writing this out, and it will help
01:56us build some intuition.
01:58We'll start by drawing two diagonal lines.
02:01I'm building like a pyramid, a pyramid of prime numbers.
02:04All right.
02:05So we're going to start at the top.
02:06We're going to write down 2, then we'll write down 3, 5, and this is going to be a good test
02:12of how well I know my primes.
02:14Then we'll do the same on the other side.
02:16Okay.
02:17And now what we're going to do is we're going to draw a line from each prime.
02:20Can you see where this is going?
02:23And where they cross is every sum?
02:26Yeah.
02:27Now I can start adding them, so we get 2 plus 2, 4, 3 plus 3, 6.
02:33You can see we're starting to get all these numbers.
02:35The main point here is just, as you go down, not only do you see all the even numbers appear,
02:40but they also seem to appear more frequently.
02:43And it seems like it should always happen, and of course the conjecture is that it does
02:47always happen, and that's known as the Goldbach's conjecture.
02:51Chen's obsession with the conjecture goes back to his high school years, when his teacher
02:55stood in front of the class telling the students all about the sciences and said,
03:00Mathematics is the queen of the sciences.
03:03Number theory is the queen's crown.
03:05And the Goldbach conjecture is the pearl on the crown.
03:09Shortly after, the teacher said, well, perhaps one day, one of you kids will solve it.
03:15And then the whole class burst out laughing.
03:18Everyone except for Chen Jim Rung.
03:22He later said, I did not laugh.
03:23I did not dare to laugh, I was worried my classmates would know my vision.
03:28But I never forgot this lesson, to always remember the pearl on the crown, and to never
03:33forget my aspirations or ideals.
03:36And now, as air raid alarms are ringing, he's maintained his vision.
03:41He's working on his math, trying to solve Goldbach's conjecture.
03:45Christian Goldbach is a little-known Prussian mathematician who grew up in the early 1700s.
03:57At first, he mostly studied medicine and law, only occasionally dabbling in mathematics.
04:02But then, in 1710, at the age of 20, he set off on a 14-year and over 4,000-kilometer-long
04:10journey to meet some of the greatest mathematicians alive.
04:14In Leipzig, he met with Gottfried Leibniz, the co-inventor of calculus.
04:18And in London, he met with both Nicolaus Bernoulli, and perhaps the greatest mathematician of
04:23all time, Sir Isaac Newton.
04:26He then finished his trip around Europe before ultimately settling down in Russia, at the
04:31newly formed St. Petersburg Academy of Sciences, where, in 1727, he met his most important
04:38connection.
04:39A 20-year-old math prodigy…
04:41Can you guess who?
04:43Leonard Euler?
04:44Leonard Euler!
04:45The two quickly became friends, bonding over a shared obsession with number theory.
04:51Then in 1729, Goldbach moved to Moscow.
04:55But the two stayed in touch, writing letters to each other for the next 35 years, literally
04:59until Goldbach died.
05:01It was in one of these letters, on the 7th of June, 1742, where Goldbach scribbled a line
05:07in the margin.
05:08It was not the exact form of the conjecture which now bears his name, but something related.
05:15It seems that every integer greater than 2 can be written as the sum of 3 primes.
05:22Euler was intrigued by the proposition, but he thought the idea could be refined further.
05:28It made sense to him that odd numbers might need 3 numbers to add together to make them,
05:33but then he turned his attention to even numbers, and he thought, well, with even numbers, perhaps
05:39just two prime numbers would be enough to sum to give you every even number.
05:44In fact, as Euler wrote in his letter, this is how Goldbach originally framed the problem
05:49in person.
05:51So Euler reformulated Goldbach's idea into two separate conjectures.
05:55The first dealt with odd numbers and said, every odd number greater than 5 can be written
06:01as the sum of 3 primes.
06:03This became known as the weak Goldbach conjecture.
06:07And the second dealt with even numbers.
06:09It said, every even number greater than 2 can be written as the sum of 2 primes.
06:15This became known as the strong Goldbach conjecture.
06:19They get their names strong and weak conjecture, because if you have the strong conjecture,
06:24then you've shown that every even number greater than 2 can be written as the sum of
06:28just 2 primes.
06:30And then you can add 3 to each of those sums to get all of the odd numbers.
06:34So if you can prove the strong conjecture, you get the weak one for free.
06:38But the reverse is not true.
06:40If you can prove the weak conjecture, you still don't get the strong one.
06:44Now after Euler reformulated these conjectures, he was so confident they were true that he
06:49wrote, I regard this as a completely certain theorem, although I cannot prove it.
06:55And he couldn't.
06:57And for the next 150 years, almost no progress is made.
07:01That is, until in 1900, German mathematician David Hilbert addressed the International Congress
07:07of Mathematicians in Paris.
07:09And there, he listed 23 of the most important problems for the 20th century.
07:14Number 8 on that list was all about prime numbers, including Goldbach's conjecture.
07:20Goldbach's conjecture was back on the radar.
07:23Only now, mathematicians started to look at the problem differently.
07:28Let's take a look at our pyramid again.
07:30One thing to notice is that this pyramid is symmetrical.
07:33When 2 primes add to make an even number on the right side, those same 2 primes make an even
07:38number on the left side.
07:39So to avoid double counting, we can look only at the right side of this center line.
07:45And what we find is that for very small even numbers, like 4, 6, or 8, well there's only
07:51one way to write them as the sum of 2 primes.
07:54But for larger numbers, like 20, well now there are multiple different ways to write it as
07:59the sum of 2 primes.
08:01And the further down you go, the more combinations you find.
08:05So mathematicians started asking a new question, not just whether every even number can be
08:10written as the sum of 2 primes, but in how many different ways it can be done.
08:15Let's call that number H of n.
08:17For example, H of 4 is 1, because you can only write 4 as the sum of primes 2 plus 2.
08:24H of 20 is 2, because that is 3 plus 17 and 7 plus 13.
08:29H of 42 is 4, and so on.
08:32Now so far, we're just counting up what the value of H should be.
08:37But in 1923, two English mathematicians, G.H. Hardy and John Littlewood, tried to come up
08:42with a function that would estimate H of n.
08:45To do it, they relied on one of the most important results from number theory, the prime number
08:49theorem.
08:51This tells you that, on average, a large number in the neighborhood of n has about 1 over
08:56the natural logarithm of n chance of being prime.
08:59So here's a crude version of what Hardy and Littlewood did.
09:03They imagined this large, even number, and they set it equal to 2n.
09:08So you could split it right down the middle at n.
09:11Now they considered a pair of numbers a and b that together add up to 2n, where a is smaller
09:16than n and b is larger.
09:19They both must differ from n by the same amount, let's call it c.
09:24Now according to the prime number theorem, the chance that a is prime is 1 over the natural
09:28logarithm of n minus c, and the chance that b is prime is 1 over the natural logarithm
09:34of n plus c.
09:35But here's the trick.
09:38Most of the times we're talking about n's that are ginormous, that are super, super large.
09:44And if you look at 1 over l and n, that function would be something like this, where it basically
09:51drops very quickly at the start, and for large numbers it's almost constant.
09:57So that this c really doesn't affect it a whole lot.
10:00So for most of the numbers, we can sort of ignore the c in both cases, and it becomes much
10:06simpler, which is just the chance of a being prime is about 1 over l and n, and the chance
10:12of b being prime is about 1 over l and n.
10:16Now what's the chance that both are prime?
10:18You're going to multiply them together.
10:20We're going to multiply them together.
10:22Still, so far we've just looked at one pair, while in total there are about n possible pairs,
10:28n ways to add up two numbers to get to 2n.
10:31So we can just multiply the chance of one pair being prime with the total number of pairs,
10:36to find the total expected number of prime pairs, which comes out to about n over ln n
10:42squared.
10:43And if we plot this, we find that the expected number of ways to write any number as the sum
10:48of two primes increases for larger and larger numbers.
10:51Hardy and Littlewood, they refine this a bit more, but essentially what they find, you
10:55know, is the exact same thing.
10:57They just have like a correction factor in front, but then this term is still the same.
11:02The only problem is that this is just an estimate, not a proof.
11:05And as they point out in their own conclusion, it is only proof that counts.
11:11And that's about as far as they get for the strong Goldbach conjecture.
11:14But on the weak one, they have more success.
11:17It all comes back to a curious incident 10 years earlier in 1913.
11:22We've invited Elmer from the YouTube channel Fern to tell you about it.
11:26Hardy sits at his breakfast table when a strange letter arrives.
11:30It's sent by a completely unknown mathematician from India, Sri Nivasa Ramanujan.
11:35The letter reads, I am now about 23 years of age.
11:39I have no university education, but I have undergone the ordinary school course.
11:44After leaving school, I have been employing the spare time at my disposal to work at mathematics.
11:49Ramanujan writes that he is a poor clerk, earning only about 20 pounds a year.
11:53He has attached some formulas and is asking for Hardy's opinion.
11:56The letter is over 10 pages long and is packed with over 100 theorems, many of them highly advanced.
12:03Yet Ramanujan provides no proofs, and very little explanation.
12:07Hardy is astounded.
12:09He has never seen anything like it.
12:11Some are already known.
12:14Others are completely new, unknown to the world.
12:17And then there are results that seem impossible.
12:19Like that the sum of all positive integers up to infinity is negative 112.
12:24How did he come up with this?
12:27Hardy shows the letter to Littlewood.
12:29The two stay up until midnight, going through the manuscript, debating whether the mysterious author is a genius or a fraud.
12:36But it sounded like Hardy felt these couldn't be crank results, because you couldn't imagine them if you were just a crank.
12:43They're too fantastical to be the work of that kind of second-rate mind.
12:49Only a really phenomenal, off-the-chart genius could have come up with them.
12:53So Hardy writes back, the rigorous and strict mathematician he is, he asks for proof.
12:59Ramanujan, however, is nothing like Hardy.
13:02He works on intuition, often guided by dreams.
13:05He would just see the result, or in some mystical versions of his own telling, he would say that his personal goddess Namagiri would come to him in his dreams and write the formulas on his tongue.
13:18Wait, on his tongue?
13:19On his tongue is the version I've sometimes heard, but in any case, he planted it in his mind.
13:25An equation for me has no meaning, unless it expresses a thought of God.
13:30Ramanujan never really learned how to provide proof to his theorems, so Hardy urges him to come to Cambridge to explain them in more detail.
13:38But there's a problem.
13:40Ramanujan is a devout Hindu.
13:42His faith forbids him from crossing the sea.
13:46Then, one night, his mother has a dream.
13:48A goddess appears and tells her to let Ramanujan fulfill his destiny.
13:53Convinced by the vision, she gives her blessing, and so Ramanujan finally sets out for England.
13:59In Cambridge, Hardy and Ramanujan start working together.
14:02I mean, I'm going to get choked up thinking about it.
14:04It's such a beautiful sentiment.
14:06I'm trying to remember how hard he put it.
14:09It's something like he thought, you know, on a scale from zero to a hundred, you know, maybe he's a 10.
14:18Littlewood is a 30, like Ramanujan is an 80.
14:22Some kind of crazy thing like that.
14:25So he knew that Ramanujan was special.
14:29Mmm.
14:30Didn't expect to get emotional for you, but this is what I do when I think about Ramanujan.
14:35But his new life in England is harsh.
14:37He's not used to the climate.
14:39He's not used to the food.
14:40And war is ravaging.
14:42It will become one of the darkest chapters of his life.
14:45If you want to know more about Ramanujan's struggles and what made him such a genius,
14:50there's an upcoming video on our channel, Fern.
14:53While Ramanujan struggled, Hardy and he did make a breakthrough.
15:01Around 1917, they invented what's known as the circle method and used it to tackle different
15:05problems in number theory.
15:07Hardy and Littlewood later developed this idea further, and for the next hundred years,
15:12this would be the main method to tackle the weak Golbach conjecture.
15:17But let's remind ourselves of the problem.
15:19We're trying to prove that we can always write an odd number, large odd number, as the sum
15:27of three primes.
15:29Call it p1, p2, and p3.
15:33But just like before, we don't only want to know whether it's possible, but in how many
15:37different ways it can be done.
15:39So Hardy and Littlewood imagined a counting machine of sorts.
15:43As input, it would take all possible combinations of three prime numbers smaller than n.
15:49For each combination, it would add them up and check if that sum is equal to n.
15:54And if it is, the machine increments a counter by one.
15:58But if not, the machine does nothing.
15:59It goes on to the next combination.
16:02Let's do an example.
16:04Say n is 11.
16:05Well the machine first checks the primes 2, 2, and 2, adds them up to get 6, which is
16:10not equal to 11, so the counter stays at 0.
16:14Next it tries 2, 2, 3, which is 7, so again it does nothing.
16:18And it keeps cycling through until it hits 2, 2, 7, which is equal to 11, so now the machine
16:24increments the counter by one.
16:27As it keeps running through primes, the counter increments only one more time for 3, 3, 5.
16:33So in total there are two ways to write 11 as the sum of three primes.
16:38But how would you go about building this machine out of mathematics?
16:43Because that is exactly what Hardy and Littlewood did.
16:47Their technique is so ingenious, I want to take you through it step by step.
16:52It starts with the function e to the i theta.
16:55This function just traces out a unit circle in the complex plane.
16:59The angle theta determines how far around the circle you are.
17:03Now imagine that theta is restricted to be some multiple m of 2 pi, and since 2 pi is one
17:09complete revolution, what doesn't matter if m is 0, 1, 2, 3, or any integer, a vector from
17:16the origin would always point in the same direction, to the right.
17:20But now Hardy and Littlewood multiply the angle by alpha, where alpha is just some number between
17:260 and 1, you can think of it as a slider.
17:30So if m equals 1 and you increase alpha from 0, well the vector slowly makes one full rotation.
17:38If m equals 2, the vector will move around faster and make two full rotations.
17:43And this keeps going for higher values of m.
17:46Now imagine taking all possible values of alpha simultaneously, and then averaging out
17:52all of those vectors.
17:54Mathematically, we can write this as the integral from 0 to 1 of e to the i 2 pi alpha d alpha.
18:02So what happens?
18:03Well, this vector cancels with this one on the opposite side, and this one with that
18:09one, and the same thing happens with all the vectors.
18:13So averaging them, we get 0.
18:16The same thing happens if m equals 2 or 3 or 4.
18:20This integral will give 0 for all values of m, except a single case.
18:27If m equals 0, then we have e to the 0, which equals 1.
18:31So no matter what alpha is, the vector always points to the right, and averaging that will
18:36always give 1.
18:39So this equation is very close to the machine we were trying to create.
18:43If m equals 0, it returns 1, but if m is not equal to 0, it returns 0.
18:49So all that's left to do is replace m with what we're interested in.
18:53Specifically, is the sum of p1 plus p2 plus p3 equal to n?
18:58We can rearrange that to get p1 plus p2 plus p3 minus n equals 0.
19:04So if we sub in this expression for m, then every time the three primes add up to n, this
19:10value will be 0, and the integral will be equal to 1.
19:14And every time the primes don't add up to n, this value will be non-zero, and so the integral
19:19will return 0.
19:21But right now, this equation only checks one triplet of primes.
19:25To check all possible combinations of primes smaller than n, we need to add a sum.
19:31Since this will count up all the ways 3 primes can add to make n, we can call it h .
19:36So in fact, let's try it out.
19:39Put in n equals 11, compute this function, and it gives 6, which is not the 2 we found
19:44before.
19:45So what happened?
19:46Well, we've simplified the logic a little bit, so we actually get some duplicates this
19:50time.
19:51For example, the machine finds 227, but also 722 and 272.
19:56Mathematicians correct for this when doing the real calculation, but for our purposes, this
20:00machine is good enough.
20:02So we've got our h , and if we can show that h is at least 1 for every odd number
20:08greater than 5, then we've proven the weak Goldbach conjecture.
20:13But there are two issues with this approach.
20:16The first is that we assumed that we know all of the prime numbers smaller than n, which
20:21is true when n is small, but as n goes to infinity, we just don't know all of the
20:26primes.
20:27The second problem is that as the number of primes increases, the number of possible combinations
20:32explodes.
20:33If n is 10,001, there are 1,229 smaller prime numbers, which means a total of 310,144,295
20:44possible prime triplets.
20:45Increase n to a billion, and the possible combinations jumps up to nearly 22 sextillion.
20:51And this trend continues, so using this approach in a simple way won't work.
20:58So Hardy and Littlewood came up with a smarter way.
21:03They fundamentally reframed the problem by taking the sum inside the integral.
21:07Rather than checking all possible combinations of prime numbers by brute force, they now analyzed
21:12the collective behavior of all the primes at once.
21:16To see how this works, take a look at the equation.
21:20In the exponent is the sum of 3 primes minus n.
21:24But adding in the exponent is the same as multiplying exponentials together.
21:29So we can rewrite this as the product of 4 exponentials.
21:33The first 3 of these are actually identical, because there's nothing to distinguish what
21:38we call p1, p2, p3.
21:40They're all prime numbers drawn from the same set.
21:43So we could just call this function s of alpha and n, and we could multiply it by itself
21:48three times, or just cube it.
21:51By doing this, h of n is now a function of just two things, e to the minus i 2 pi n alpha,
21:58which is just a complex number that you could compute, and this new function s of alpha and
22:03n.
22:05But what is this function?
22:07Well, let's do an example.
22:09Again, say n equals 11.
22:12Then s of alpha and 11 equals just a sum of 4 exponentials, each governed by its own prime
22:19number.
22:20You can think of each of these exponentials as a clock, each with its own prime number
22:24that determines how fast the clock spins.
22:27So 2 turns the slowest, 3 turns faster, 5 turns faster still, and so on.
22:32But remember, this is a sum, so we actually need to add up all the clocks tip to tail.
22:38Now watch what happens as we increase alpha.
22:40All the clocks wind around at their own rates.
22:43What we're interested in is not the behavior of any individual clock, but instead all of
22:48them together.
22:50Since we've chosen a pretty small value for n, nothing too exciting happens.
22:54But watch what happens if we pick a larger value for n, like 99.
22:59Now there are 25 prime numbers that are smaller than 99, and hence 25 clocks in our sum.
23:06Now as we increase alpha, all the clocks again wind around at their different rates.
23:11It almost looks like a dragon's tail.
23:13And for a while, everything seems kind of chaotic, circling around the origin.
23:18But then, when alpha hits 1 over 6, the tail unwinds.
23:23And suddenly, almost all the clocks point in a similar direction.
23:27And we get a large resultant value.
23:29If we increase alpha further, the clocks cancel out again, until we hit the next magic point,
23:351 over 3.
23:36Here again, many clocks line up.
23:38And this curious pattern keeps repeating.
23:41For the majority of values of alpha, the clocks cancel each other out.
23:45But at a few special points, like a half, the tail unwinds, the clocks interfere constructively,
23:52and we get a large resultant value.
23:54If the angle is, you know, 360 degrees divided by a whole number, you get some very particular
24:00behavior that's very predictable.
24:02Those are the angles at which this function, when you sum the primes, they actually go far
24:07from zero.
24:08And what happens at a random angle, and you just start adding the primes up, it keeps
24:13changing which way it's pointing, and the thing just bounces around.
24:17So why is this happening?
24:19Well, let's look at when alpha equals a half.
24:22Here, what we're doing is multiplying each prime number in the exponent by a half.
24:27In other words, dividing each prime number by two.
24:30So let's do this for each of the primes.
24:32In the first column, we'll write down how many times it can be divided by two, and in
24:37the second column, we'll write down the remainder.
24:40So two divided by two equals one, leaving a remainder of zero, so we've just made one
24:45full rotation.
24:47Three divided by two is one, but that leaves a remainder of one, so it makes one full rotation
24:52and then a half rotation.
24:54Five divided by two is two with a remainder of one, so it makes two full rotations and then
24:59a half rotation more.
25:01Now, we don't really care about the first column because it doesn't matter how many
25:05full rotations a clock makes.
25:07We only care about where the arrow ends up, and that's fully determined by the remainder.
25:12It's just the remainder times alpha.
25:14So from now on, we'll just keep track of the remainders, and what we find is that all
25:18of the other remainders are equal to one.
25:20So all clocks except the first one point the same way.
25:24This is because when we're dividing whole numbers by two, there are only two possible remainders,
25:29zero, or one, and since we're working with prime numbers, only two can have the remainder
25:33zero because no other prime number is even.
25:37Therefore all other primes must share the remainder one, meaning all those clocks end
25:42up making another half rotation, so they all point left and constructively interfere.
25:47A similar thing happens when alpha equals a third.
25:50Now each prime is divided by three, so the possible remainders are zero, one, or two.
25:55Again, three is the only clock that has remainder zero, so that point's right, but all the other
26:00primes follow a special pattern, a pattern that makes the entire circle method possible.
26:06The remaining primes distribute themselves roughly equally between remainders one and two,
26:11which means all the remaining clocks are rotated one-third or two-thirds of the way around the circle.
26:16Since these clocks are pointing to the left, again we get constructive interference.
26:22And this pattern of constructive interference also occurs for other small rational fractions.
26:27But for non-rational numbers, you don't get these nicely restricted remainders, so the
26:32clocks are roughly distributed equally all over the circle, and you get destructive interference.
26:38Hardy and Littlewood realized they could use this.
26:41Imagine taking this plot of the magnitude of the resultant arrow and wrapping it around
26:46a circle.
26:47What they found is that the majority of the contributions come from very small regions of the circle,
26:53what they called the major arcs.
26:55The rest of the circle only adds minor contributions, so they called these the minor arcs.
27:01This allowed them to split up their calculation into two parts.
27:05The major arc part gives you the main term for how many ways you can write a number n as the
27:10sum of three primes, and the minor arcs just gives you an error term.
27:16They then showed that if the generalized Riemann hypothesis is true, then the main term grows
27:20faster than the error term.
27:22So eventually for some large enough number, call it k, this value will always be larger than
27:27one, and the weak Goldbach conjecture holds.
27:31But there are two problems with this approach.
27:34The first is that they assumed that a generalization of the Riemann hypothesis is true, which we
27:39don't know.
27:40And the second is that they didn't actually specify that number, k.
27:44I mean, how big of a number do you need to get to before you can guarantee that all numbers
27:49larger than that adhere to Goldbach's conjecture?
27:53Then we've got to wait until 1937, when Russian mathematician Ivan Vinogradov proves the same
27:59as Hardy and Littlewood.
28:01But he did it without the generalized Riemann hypothesis.
28:03So, you know, assumption three.
28:05It's one thing to just assume the Riemann hypothesis and say, oh yeah, you can make some progress.
28:11But Vinogradov's like, no guys, you don't need the Riemann hypothesis, you can really prove
28:14this stuff.
28:15But again, he gives some large enough number for which the weak Goldbach conjecture will hold,
28:19without specifying that number, and so it's very unsatisfying.
28:24Over the next 19 years, one of his students did get that satisfaction when he could specify
28:28that number.
28:29Do you want to have any guess as to what the number is?
28:3210 to the 50, 10 to the 80, 10 to the 3000, I don't know.
28:37I was guessing things that are far too low, probably.
28:39Yeah, yeah, yeah.
28:40It's about 10 to the 6.8 million.
28:44Wow, okay.
28:46It's huge.
28:47Even, you know, from here on, people start working on his method more, and they all use
28:51the same techniques.
28:53In 1989, the number was brought down to 10 to the 43,000.
28:57Next, it dropped to 10 to the 7,194.
29:02And by 2002, it had dropped all the way down to 10 to the 1,346.
29:06Now, that seems so much smaller than what we had before, but if you compare it to how many,
29:14you know, protons there are in the universe, which is 10 to the power 80, that's not something
29:18you can check by computer.
29:19It's absolutely, absolutely hopeless.
29:21Even if you turned the whole universe into a computer, you would still have no chance of
29:25solving this.
29:26So it really needed to get dragged down.
29:30This is roughly where the problem stood in 2005, when Peruvian mathematician Harold
29:34Helfgott became interested in it.
29:36He was reading papers about the Wiggoldbach conjecture and new techniques that were being
29:42developed.
29:43So these were the cutting-edge techniques at the time.
29:45He was reading it, and he thought, I can do better.
29:48I'm having some ideas of how to do better, and I, you know, it took me a couple of weeks
29:52to get better estimates, not still, you know, strong enough to revolutionize everything,
29:58but to make real progress.
30:00So he gets to work, and within a couple of weeks, he's got better estimates than that
30:04in the paper.
30:05So now he's getting more confident, like, okay, I got this.
30:09Of course I was a bit naive.
30:11I underestimated some difficulties elsewhere.
30:14To solve the problem, he realized he needed to do two things.
30:17One, get the computational number as high as possible.
30:22And two, get the constant down as much as he could, which is where the difficult math
30:27comes in.
30:28So over the next eight years, he attacked the problem from both sides.
30:32Together with David Platt, he checked all numbers up to 8.8 times 10 to the 30 by computer.
30:39And by refining his math, he slowly brought down the constant number k.
30:45By 2013, he gets the number all the way down to 10 to the 27, which is below the number
30:52that his computers had already checked.
30:55So he solved it.
30:56He wrote up his results in a paper called The Ternary Goldbach Conjecture is True, which
31:01is quite the mic drop title.
31:03So, nearly 300 years after Goldbach's original letter, Havgott proved that every odd number
31:09greater than 5 can be written as the sum of 3 primes.
31:13And as a result, he also proved that every even number greater than 2 can be written as the
31:18sum of at most 4 primes.
31:20This is because you can always just add 3 to all the odd numbers to get all the even numbers.
31:26So was it a big deal?
31:27Were people excited?
31:28Yeah, it's a good question.
31:30I feel like it is mostly an obscure problem.
31:33I think the strong one, and this with both of these problems, they don't have any direct
31:38application to the real world.
31:40Because it's so simple, it has this massive wide appeal and everyone will kind of know
31:43about it.
31:44It's also one of the oldest unsolved problems.
31:47What did you do when you proved it?
31:49Well, I put it in the archive, like any normal person, it was a big relief.
31:55Given your success with the weak form of the Goldbach's conjecture, how do you feel about
32:00the strong form?
32:01Oh, I think that's hopeless for the moment, and if it yields, if it were to yield during
32:06our lifetime, I wouldn't bet on it, in fact I would bet against it.
32:11See the reason the circle method works for the weak Goldbach conjecture is because this
32:15main term grows faster than this error term.
32:18But that's not the case for the strong Goldbach conjecture.
32:21The major arcs are no longer major.
32:24The main contribution comes from the minor arcs, or at least as much of the contribution
32:29comes from the minor arcs.
32:31So to solve the strong conjecture, we need some fundamentally new technique or approach.
32:36But we haven't found it yet.
32:38The person who arguably got the closest was Chen Jing-run.
32:43In 1956, Chen was recognized for his mathematical prowess, and a year later he became an assistant
32:50at the Chinese Academy of Sciences, where he spent the next ten years working on problems
32:54in number theory, including Goldbach's conjecture.
32:58By 1966, he hit a major breakthrough.
33:01By using another approach known as sieve methods, he proved that every sufficiently large even
33:07number is the sum of a prime and a number that's either a prime, or the product of exactly
33:12two primes, what we call a semi-prime.
33:15It's a very learned and brilliant proof.
33:18I think André Wey compared following his proof is like climbing along the top of the Himalayas.
33:24You know, that you're way up high in practically the stratosphere.
33:29It's such a sophisticated argument.
33:31This was the closest anyone had gotten to solving the problem.
33:35Elated, Chen showed the proof to a colleague, who suggested he should announce the result and
33:40then tidy it up before publishing.
33:42But just as Chen announced his result, the country plunged into chaos.
33:48In May 1966, Chairman of the Communist Party, Mao Zedong, declared that bourgeois elements
33:55had infiltrated the government and society, and they must be purged.
33:59This marked the start of the Cultural Revolution, a radical campaign to defeat perceived enemies
34:04of the party.
34:06With support from Mao, militant student revolutionaries known as the Red Guard stormed institutions.
34:11They burned books and turned on intellectuals.
34:14Teachers, scientists, and professors were subjected to brutal public spectacles known as struggle
34:20sessions.
34:21They were beaten, humiliated, and forced to wear towering caps and placards listing their
34:26supposed crimes, like capitalist sympathizer or enemy of the revolution.
34:32Across China, the pursuit of knowledge was replaced by fear and political conflict.
34:37By 1968, 131 of the 171 senior members of the Chinese Academy of Sciences in Beijing faced
34:44persecution, and many others lost their lives.
34:47Chen was one of the targets.
34:50He was forced into manual labor and made to live in a converted boiler room with no electricity.
34:55The Red Guard insulted him, spat on him, and beat him so badly that Chen often lost consciousness.
35:02In one report, it was so bad that he may have tried to commit suicide.
35:07He jumped off of a third-story building and fortunately didn't hit the ground.
35:15He landed on the second floor and got injured but didn't die.
35:21Despite all of this, Chen kept secretly working on his math under the dim light of a kerosene
35:28lamp.
35:29Then, on the 13th of September, 1971, Mao's right-hand man died in a mysterious plane crash
35:35after supposedly trying to organize a coup.
35:39His death shattered the illusion of unity in the party, and the Cultural Revolution began
35:44to lose steam.
35:47But Chen faced a dilemma.
35:48On the one hand, he wanted to publish his proof.
35:51But on the other, he was terrified of being criticized.
35:54Unsure what to do, he turned to Luo Shengchong, the head of the Mathematical Institute, who told
35:59him, the result is sound.
36:02You must publish.
36:05And so, in April 1973, Chen published his theorem.
36:10Three years later, Mao Zedong died and the Cultural Revolution ended.
36:16Famous journalist Zhu Qi took note of Chen's story and wrote an article about his achievements
36:21and hardship.
36:22The article was reprinted in China's most popular newspaper and read by millions.
36:27The state that just ten years earlier silenced Chen now celebrated him as a national hero.
36:33His story was retold in schools, books, and films.
36:38He even got an asteroid named after him in 1996.
36:40And imagine if he hadn't done it, you know, if he hadn't published it.
36:46Like all his work, his whole mission, his whole ideals, his whole vision, maybe no one would
36:50have known about it.
36:51It would have just been a little line somewhere, like, oh, I've got this proof, but yeah, I'm
36:54not going to show it to you, you know.
36:57But so often we, like, shoot ourselves down, whereas really do your best and then put it out
37:02into the world and then let the world sort of, like, evaluate it.
37:06And while Chen got incredibly close, he was never able to fulfill his ultimate dream, to
37:11solve the strong Goldbach conjecture.
37:14Perhaps one reason why it's so hard to prove true is because maybe it's not, maybe it's
37:19false.
37:20But if it is false, then there's a very easy way to show that, by finding a counterexample,
37:25a single even number that can't be written as the sum of two primes.
37:30If Goldbach is false, it would be very easy for you to convince me that Goldbach is false.
37:35Just find a counterexample.
37:36That is the only way that you can convince me that Goldbach is false.
37:39If Goldbach is false, then it's false because there's a counterexample.
37:42And if there's a counterexample, then you can give me the counterexample and I can check
37:45for myself that yes, that's a counterexample.
37:47In this pursuit, in 1938, a mathematician named Nils Pipping checked all numbers up to 100,000
37:53by hand, but he didn't find a single counterexample.
37:58Since then, modern computers have brought this number up to four quintillion, and the conjecture
38:03holds for every single one of them.
38:07In fact, we can also use computers to check in just how many ways a different number can
38:11be written as the sum of two primes, which gives us this pattern.
38:16It looks like a curve with tails.
38:18It strongly resembles a comet, which is why this is known as Goldbach's Comet.
38:24It shows that as the number gets larger, there are more and more ways to write every even
38:29number as the sum of two primes.
38:31And if we overlay our heuristic argument from before, we find that it matches up remarkably
38:36well.
38:37The one way the conjecture could be false is if, for whatever reason, at some very large
38:43number, there's a conspiracy where this completely breaks down and it suddenly drops below that
38:48line.
38:49Are there any examples where you suddenly have a big drop in the number of ways you can
38:53compose a number?
38:54No, no.
38:55Those asymptotics seem to hold very nicely.
38:57There are no big drops.
38:59So it seems like the conjecture should be true.
39:02We've tested by brute force all the numbers up to 4 quintillion, and they all obey Goldbach's
39:08conjecture.
39:09So you might think that we should have found a counterexample by now, or at least a number
39:13that doesn't follow this general trend we've seen.
39:16But on the scale of all numbers, 4 quintillion is nothing.
39:20So we'll have to come up with some smart approach, or fundamentally new way of solving this problem.
39:26Now, as far as we know, solving Goldbach's conjecture doesn't really affect any other areas
39:31of math.
39:32And it also doesn't seem to have any direct applications to the real world.
39:36So you might think, why bother solving this problem?
39:39Why not work on some big problem that we know is important?
39:43But I think that misses the point.
39:45What if the Goldbach conjecture is, turns out it is opening up a whole new part of the
39:50multiverse of mathematical ideas.
39:52We don't really know, because we don't have the bird's eye view or the god's eye view of
39:56math.
39:57It's what it looks like to us right now on the web of math.
40:00So I tend to dislike this picture of certain things are central and other things are peripheral.
40:07People should do what fires them up.
40:09Because if you do that, you'll be passionate, you'll think about it all the time, you'll do
40:14it when you're in the shower, you'll think about it when you're driving.
40:18And you might do something remarkable because of that passion.
40:21And if you're just doing something because you think it's important, I think you'll
40:25tend to be second rate, honestly.
40:28I like the modest view, that we don't know what's necessarily important, but we do know
40:33what we love.
40:35So work on that.
40:41Goldbach's conjecture frustrated mathematicians for centuries.
40:45And for a long time, it seemed like an unsolvable problem.
40:49Many gave up, thinking it was just too big to tackle.
40:52But as we've seen, it only takes a few incredibly determined individuals who reject the status
40:57quo to keep pushing towards a solution.
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