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  • (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 innigsberg, 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...

(Speaking German)

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ゴールドバッハの思い込み - Numberphile (Goldbach Conjecture - Numberphile)

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    林宜悉 に公開 2021 年 01 月 14 日
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