(Speaking German)
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...