Lesson video

Hi, I'm Miss Kidd-Rossiter, and I'm going to be taking this really exciting lesson on Goldbach's conjecture.

This is an as yet unsolved maths problem, that's been causing mathematicians all over the world a bit of a headache for a long, long time.

So I'm really excited to get started with it.

Before we get going you're going to need a pen and something to write on.

If you've got any large paper A3 or bigger, that might be helpful for today's lesson.

Make sure you're in a nice quiet place and you're free from all distractions.

If you need to pause the video now, to sort anything out, then please do.

If not, let's get going.

Right, let's get started.

So with the try this activity, Zaki is thinking about prime numbers.

He says any number can be written as the sum of two prime numbers.

What does sum mean? Tell me now.

Excellent.

It means that we're adding doesn't it.

So that's what he's saying.

Well do you agree with Zaki? And can you find some examples that either show or don't show this to be true? Pause the video now and have a go at this task.

Excellent.

Did you agree? Tell me.

Okay, brilliant.

So can you tell me an example that shows that we can write a number of the sum of two prime numbers? Excellent.

Okay, we could write seven as two plus five.

Two is prime.

Five is prime.

So we can write seven as the sum of two prime numbers.

Can we write two as the sum of two prime numbers? No, we can't.

So that is one that doesn't show this to be true.

What about one? Excellent.

Another one that doesn't show this to be true.

So Goldbach's conjecture is all to do with writing numbers as the sum of two primes.

So Christian Goldbach was born in 1690 and he died in 1764 and he is most famous for this conjecture that he wrote in a letter to Leonard Euler.

And he's a really, really famous mathematician.

So on the 7th of June, 1742, he wrote this: Every even integer greater than two can be written as the sum of two prime numbers.

So let me read that to you again: Every even integer greater than two, can be written as the sum of two prime numbers.

So let's just, uh, pick this apart a little bit.

So integer, what does that mean? Excellent, a whole number.

So every whole number greater than two can be written as the sum.

Remind me what that means again? Excellent.

Adding two prime integers.

And again, we've got the word integer, which means whole numbers.

Let's just recap what's a prime? Excellent, a number with two factors.

Exactly two factors I should say.

And what are those two factors? Excellent.

Itself and one.

So every whole number greater than two that's even can be written by adding two prime numbers together.

Prime whole numbers.

So this is thought to be true.

We've not managed to disprove this statement yet, but we've also not managed to prove this statement.

So that's why it's one of the really troubling, unsolved problems in maths.

And you can see how old it is.

One way to represent this conjecture is using a very common diagram.

So I'm going to draw an arrow or a line that would extend further.

I'm going to put all my numbers on it greater than two.

So greater than or equal to two, I should say.

And then I'm going to do the same on the other side.

And then from each of these numbers, draw a parallel line to the opposite line.

So like this, and then also for the other one, like this.

Now I want to remove all the numbers that are not prime.

So two is prime.

Three is prime.

Four is not prime, so I'm going to remove that and its line.

Six is not prime, so I'm going to remove that and its line.

And you could continue this diagram even further.

So that's why I've asked you to get some A3 paper, or even A4 would be fine, if you want to have a go at continuing this diagram.

So let's remove the numbers that are not prime.

And now we can see that any place that these two lines intersect, it is the sum of those two prime numbers.

So for example, here, we've got two plus two.

So this intersection represents four.

So we can write four as the sum of two prime numbers.

What does this intersection here represent? Tell me now.

Excellent.

Five.

What about this intersection here? Excellent.

Seven.

So we've got some odd numbers at the moment.

What about this intersection here? Excellent.

Nine.

And then we can see can't we, that we're going to have to repeat some of these going down this side.

What would this intersection here be? Excellent.

Six.

What would this intersection here be? Eight.

Good.

What would this intersection here be? 10.

Good.

What would this intersection here be? 10 as well.

So you can see that some numbers we can write as the sum of two prime numbers in two different ways or more different ways.

What about here? Exactly.

This is eight.

This one is 10.

This one here is 12.

Good.

And this one here is 12.

Excellent.

Have I missed any out on that diagram? I can't see that I have.

So we can see that we've got all the even numbers there greater than two and less than or equal to 12 represented by adding two prime numbers.

So we've got four, six, eight, 10, and 12.

And as I said, you can continue this diagram as far as you like and see how many different even integers you can write as the sum of two prime numbers.

You're now going to apply your learning to the independent task.

So pause the video now, navigate to the independent task and then have a go at answering the questions.

When you're ready to go through some of them and discuss them, not all of them will have one correct answer, then resume the video.

Good luck.

How did you get on with that? Some tricky questions there and some really interesting things for you to think about as well.

So list all the prime numbers up to a hundred.

That one should have been fairly straight forward.

There you go.

Then I asked you to continue the diagram.

What was the largest even number you could write as the sum of two prime numbers? Now, every single one of you is going to have a different answer for this, but let me tell you something really interesting.

The Goldbach conjecture has been shown to hold, so that means it is true, for every number up to and including, every even number I should say, up to and including four times 10 to the power of 18.

So that is four with 18 zeros.

Can you tell me what that number is? Go and research it if you don't know.

What do you notice? Did you make any conjectures? I'd be really interested to hear them if you did.

Explain why the conjecture cannot hold for most odd numbers.

So when we add two prime numbers together, why can it not hold for most odd numbers? Well, we know that for an odd number, if we're adding two numbers together, to get an odd number, we have to add an odd and an even don't we.

If we add two even numbers together, we get an even number.

If we add two odd numbers together, we get an even number.

So the only even prime number is two.

So if we cannot form the odd number by adding two to another prime, then it cannot hold can it? So for example, 19 we could write as a sum of two prime numbers, because two is prime and 17 is prime.

However, 11 we couldn't write as a sum of two prime numbers because we would have to write it as two plus nine, but nine is not prime.

Because it has factor of three as well, doesn't it? So that's why it doesn't hold.

Thinking about the explorer task now, Yasmin is thinking about Goldbach's conjecture.

She says, as even numbers get larger, then there is more ways to write them as a sum of two prime numbers.

Do you agree with Yasmin? Can you find some examples that either show or don't show this to be true? So pause the video now and have a go at this task.

How did you get on with that? Well, the thinking is that this is generally true.

So we could think about, first of all, we can write 12 as the sum of seven and five, but we could write 10 as the sum of five and five.

And we could write 10 as the sum of seven and three.

So you can see that there's only one way for 12, but there's two ways for 10.

This is interesting, isn't it? But as we get larger, are there more ways, for example, to write 1000 as the sum of two primes, then there are to write 100 as the sum of two primes.

That's something interesting for me to leave you with, to think about, and if you're really interested in this, then it might be worth researching something called Goldbach's comment.

And that's a really nice representation of this statement here.

So that's the end of today's lesson.

There is absolutely loads of research going on in the mathematical community around this conjecture.

And so there are so many mathematicians that would absolutely love to be able to prove this.

Other mathematicians have done some research around the conjecture and some different things.

So for example, in 2013, there was a paper published that proves that every odd number can be written as the sum of.

I'm not telling you how many primes.

I'll let you research it.

But I hope that you've found this really interesting and that you have learned a lot about something quite tricky in the mathematical community that has stumped mathematicians for a long time.

If you'd like to share your work with Oak National, then please ask your parent or carer to share your work on Twitter, tagging @OakNational and #LearnwithOak.

Hopefully I'll see you again soon.

Bye!.