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  • I have the number 210. Let's try to factor this into prime numbers.

  • So 210 is 2 times 3 times 5 times 7.

  • It's the product of the first four prime numbers. This is its unique prime factorization.

  • And what are these prime numbers? These are numbers that can't be factored into smaller numbers.

  • And we leave 1 out of the game because it's special. And so the prime numbers are 2, 3, 5, 7, 11, 13, 17, 19

  • and they go on forever, as was proved

  • maybe 2,300 years ago by Euclid. Now one thing that you can

  • obviously see about these primes is that, after 2,

  • they're all odd numbers.

  • That means that, except for 2, if you add two primes together you get an even number.

  • So the two primes could even be the same prime, so for example:

  • I get 3 plus 3 equals 6.

  • I get 3 plus 5 equals 8.

  • I get -- let's see -- 3 plus 7 equals 10.

  • I also have that the same as 5 plus 5. So 10 has two representations as a sum of two primes.

  • There's a conjecture of

  • Goldbach from the 18th century that every even number starting with 6 is the sum of two

  • odd primes. And I should state this with a question mark.

  • It's -- It's -- It's not known. There have been novels written about the quest for proving this conjecture of Goldbach.

  • We don't know the answer.

  • My friends who are analytic number theorists would die to prove Goldbach's Conjecture.

  • It really is, would be a great prize, a great coup.

  • We strongly suspect that the answer is yes, and why do we suspect so?

  • Well, because the primes are very numerous. Even though they thin out,

  • they are very numerous. And so, a given even number has a lot of chances to be the sum of two primes.

  • 210, as you can quickly see, is an even number. And Goldbach would predict that it's the sum of two primes.

  • In fact, it's not only the sum of two primes. It's the sum of two primes in a lot of ways and a champion

  • number of ways. So what I mean by that if -- if you're going to write 210 as p plus q

  • where p is less than or equal to q and p and q are prime numbers and q Is

  • greater than or equal to 105.

  • So let's take -- let q be any prime with 105

  • less than or equal to q, less than or equal to 210. OK? Any prime whatsoever and now let's look at

  • 210 minus q. What can you tell me about that number? That number Is

  • less than or equal to 105.

  • 105 being half of 210 and q Is above the halfway. 210 minus q is not divisible by 2.

  • How do I know that?

  • Because 210 is divisible by 2 and q is a prime and it's bigger than or equal to 105 so we're taught.

  • OK?

  • It's not divisible by 3. How do I know that? Because 3 Is a divisor of 210

  • and q, being a prime and bigger than or equal to 105, Is not divisible by 3.

  • And also this number is not divisible by 5 or 7.

  • For the same reasons, since 210 Is 2 times 3 times 5 times 7.

  • So this number does not have -- as prime factors -- 2, 3, 5, & 7.

  • So what is its least possible prime factor?

  • Well, it would be the next prime after 7 which is 11.

  • So if this --

  • -- a composite number, it would have to be some prime greater than or equal to 11 times something else

  • bigger than 1. But the least candidate is 11 times 11

  • which is 121 which is bigger than 105. So we conclude that 210 minus q

  • is 1 or prime.

  • And it's not 1.

  • And how do I know it's not 1?

  • Well, if it were 1, that would mean that q is 209 and q is not equal to 209

  • because 209 is not a prime (it's 11 times 19). So every prime q

  • between 105 and 210 gives us a p such that p plus q equals 210.

  • So, Goldbach is not only right about the number 210,

  • he's REALLY right about the number 210. And -- in fact --

  • you can ask how special is this property that 210 has.

  • It's the last number with this property.

  • So let's look at all these primes between 105 and 210.

  • Here they are.

  • Look at them.

  • Every one of them is a candidate for writing 210 as a sum of two primes

  • And, if it works, then 210 minus one of these primes will itself be a prime.

  • Is there a pairing for each one of these primes where the prime's smaller than or equal to 105?

  • And we find 210 is the very last case where this ever happens.

  • It has happened for a few smaller cases so, for example, for the number 36.

  • It works. OK, that's boring. 36 is small.

  • 210 is, you know, getting up there and it works for 210,

  • OK? It doesn't work at all for any other number bigger than 210.

  • [Brady] And this is a -- this is proven?

  • [Carl Pomerance] And this is a proved theorem, and I can even tell you the names of the people that proved it.

  • It was done in 1993 and the authors were:

  • Jean-Marc Deshouillers, Andrew Granville, Wladyslaw Narkiewicz, and myself.

  • [Brady] Professor, I'm going to call 210 VERY Goldbach-y

  • [Carl Pomerance] Very Goldbach-y.

  • That's an excellent name, Brady. I like it.

  • [Professor David Eisenbud] -- 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 --

I have the number 210. Let's try to factor this into prime numbers.

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A2 初級

210是非常的哥德巴赫 - 數字愛好者。 (210 is VERY Goldbachy - Numberphile)

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    林宜悉 發佈於 2021 年 01 月 14 日
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