That means that a given number is the sum of two primes.

I hold it for a completely certain result.

Looking aside from the fact that I can't, myself, prove it.

So we're talking about Goldbach's Conjecture. One of the real old chestnuts of mathematics.

Christian Goldbach was born in 1690 in Königsberg, now part of Russia.

And he was a fairly serious mathematician.

But his great moment of fame came in a correspondence with Leonard Euler,

who was really one of the great mathematicians of all time.

And in a letter that he wrote to Euler,

on the seventh of June in 1742.

In that letter, Goldbach proposed this conjecture,

which got sort of ironed out after a few rounds.

But it is a famous conjecture now which says

that every even integer is the sum of two prime numbers.

Why even integers?

Well prime numbers are mostly odd numbers

and if you add two odd numbers you always get an even number

Let's make a picture that shows how this happens.

I'm gonna write two lines, which both have the primes on them.

3. And let's see, 4 is not a prime

5 is a prime, 6 is not a prime.

7 is a prime. 8, 9, 10, are not primes,

11 is a prime

Let's put a 2 in just to please Brady.

Because two is a prime. Why should we discriminate against two?

So here, I'll put two on the other side as well.

2 is a prime. 3 is a prime. 4 is still not a prime.

5 and we could go on

So this is an infinite game. And now let's draw lines that connect the sides.

So I'm gonna try my best to draw a line parallel

to the lines I've drawn.

And now let's look at where these lines intersect

and write down the sums of the two primes that are coming

So here's four, which is 2+2.

3+2 is 5.

Well, that's not even but we'll put it down anyway just for the sake of having something there.

And of course we have 7 and 9. Now 3+3 is the first interesting one.

Here's 6. And let's see what else do we have...

We have 5+3 is 8

And 7+3 is 10

And 11+3 is 14. That's a bit of a skip. I might be worried about that one

And 13+3 is 16

You mean because we've skipped 12?

We skipped 12? Did we get a 12? Well yes of course we do. We get 5+7.

Here's 12 and here's 7+5

This is a symmetric picture so it's not too surprising we get the same thing both ways.

7 and 7 we saw one 14 before. This is a really different 14.

We had 3+11 or 7+7.

Uh, what's this one? 11 and 5

Another 16. Here's 5+11 another 16.

And then here's 5+13 is 18. And 7+11 another 18. A really different 18.

I have 13+7=20.

Here's 11 and 11: 22. And as you can see as it goes

down I sort of fill things out. Now what

you don't see from this picture so well

is that actually as the numbers get big

there are really a lot of ways of

writing the numbers as the sum of two

primes. And in fact you can estimate how

many in a very crude, simple-minded way,

and it turns out to be that you're

really pretty close for big numbers. So

let's do a little calculation. One

of the most famous theorems in number

theory is the prime number theorem and

it says that the density of the primes

around n, so the chance of a number near n

being prime, is n divided by the natural

logarithm of n. That's the prime number

theorem. I'm not going to try to prove it

or justify it, but it's true. So using

that, we can estimate the number of ways

to write a given number n, or 2n let's say,

as the sum of two primes. Let's

use a different number, 2m. So how many

ways can you write 2 times m, that's an

even number, as p plus q where p and q

are prime?

That seems pretty mysterious. They just,

if you don't know anything about it, but

it's easy to analyze. So if you write

2m as the sum of p plus q, then one of

p and q had better be bigger than m,

bigger than or equal, and the other one

will be less than or equal to m.

If we look at a particular number that's a

little bigger than m, between m and 2m, its

chance of being prime is 1 over the log.

Now logarithm doesn't change very fast.

It's a very slowly growing function. So we can

estimate it as being the log of m. So if

I write, if I write 2m as a plus b,

where a is bigger than or equal to m and

b is less than or equal to m, then

the chance of a being prime is about 1

over the log of m. So the chance the

probability that a will just by accident

be prime is about equal to 1 over the

log of m. Okay. And that's the same for

the chance of b being prime. It's about 1

over log of m. So the chance of both of

them being prime at the same time is one over

the log of m squared. Well, that's a

bit of a fib. It would be 1 over log m

squared if they were independent events,

but it's not quite independent. We'll

talk about that in a minute.

How many chances do we get? To compute

this probability we had to choose an a

between m and 2m, so there are m choices.

Number of ways to write 2m as p plus

q is about equal to m divided by the log

of m squared. m is a whole lot bigger

than the log of m. Think of base 10.

Think m is a million and the log of m is

6. So you know, this is like a million

over 36 and if it's a billion or a 10 to

the 12th let's say, then m would be 10 to

the 12th and this would be 12. So 10 to

the 12th over 144 in other words this is

this is pretty close to m actually. So

it's an enormously large number. So for

any given large number there are gonna

be lots of ways of writing it as a

sum of two primes.

Somebody I was talking to said maybe

that's why it's so hard to prove. If there

were a unique way, then maybe you could

just find it. You could figure out the

formula for it. But if they're just any

old ways, almost everything works, then

how are you going to find that needle in

a haystack? Or the haystack around the

needle, rather? This is very heuristic,

right? We didn't prove anything in this

little discussion. We just made a guess,

but it turns out to be a very good guess

and there people who have tabulated these

things. There's something called Goldbach's

Comet. For each number m you show

the number of ways of writing it as the

sum of two primes. And it grows just as you

would expect, like this. You see this

wonderful picture. There's some variation

of course, some numbers have lots of ways,

some a few. But even the ones with the

fewest ways, the number seems to grow pretty steadily.

Brady: "Do you ever get a

"really really big even number that has

"like only one way? Or is there always lots of..."

No one has ever found such a thing, I think. It really

just keeps growing. I don't know if there's

any lower bound known or guessed.

So it remained unproven all this time.

People have proven other things, people

have made other conjectures around it.

For example, Harold Helfgott finally

in 2013 managed to prove that every odd

number is the sum of three primes, and

that actually implies that every even

number bigger than something is the sum

of four primes. So, you know, that's

something. And Hardy and Littlewood,

famous famous number theorists, decided

that it was too bad to leave out the odd

numbers, so sum of three primes, that's all

very well. But let's make it more special.

How about the sum of a prime and twice a

prime? So the sum of three primes would

take two of them to be equal. So they

conjected that that was true. Nobody can

prove that either, but all these things

seem very likely to be true. My friends

who are analytical number theorists would

die to prove Goldbach's conjecture. It

really is, would be a great prize, a great

coup. You know the professionals are

shy about them. If I talk to you

about Goldbach, you might think I'm actually working on it. You might think I'm

nuts. But, because nobody really has a

clue how to attack it, I think. But

nevertheless, people do work on it and

sometimes in their closets, sometimes in

their attics. I'm sure lots of my friends

would love to prove it, so secretly. I

swear I've never worked on Goldbach's Conjecture, honest to God.

22, 24, 22, 23 it's a lot

of information. And then one more thing,

one more thing to you we can learn from

this, a prime number formula: the nth

prime twiddles, or is approximately n

lots of...