(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...
David Eisenbud is so awesome