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 --