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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

• 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)

A2 初級

# 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