Lesson video

Hi I'm Miss Kidd-Rossiter.

And I'm going to be taking today's lesson on the twin prime conjecture.

It's a famous unsolved problem in maths.

Before we get started, you're going to need something to write with and something to write on.

If you need to pause the video now to get that sorted, then please do.

If not, let's get going.

So before I tell you what the twin prime conjecture is, I'd like you to try this activity.

You put Yasmin on your screen, and she says, I'm going to investigate the difference between consecutive prime numbers up to a hundred.

What's the smallest difference Yasmin could have found? what's the largest difference Yasmin could have found? And what do you notice about the differences? Pause the video now and have a go at this task.

How did you do on that? What was the smallest difference Yasmin could have found? Well, that exactly the smallest difference is the difference between the first two prime numbers, which is one.

So the first prime is two and the second prime is three.

And their difference there is one.

We call this the prime gap.

What was the largest difference Yasmin could have found? Well, if you look to all of the consecutive primes up to a hundred, there were lots of gaps of six, but the largest gap was actually of eight, which was between 97 and 89, which gave us a prime gap of eight.

What did you notice about the differences? We should have noticed that apart from the first prime gap, apart from the first difference, all the differences were even numbers.

Why is that pause the video.

Think about it.

Exactly.

So this counting coup whenever your finding the difference between two prime numbers, you're always doing an odd number, take away another odd number, and that gives you an even number.

So the first prime gap, which is the difference between an odd and an even number will give us an odd answer, but then all the other prime gaps going forward will always be even.

So the twin prime conjecture States, this, there are infinitely many primes, such that add two to that prime number.

And that number is also prime.

So for example, we know that 11 is prime.

If we add two to that.

So this is if P equals 11, then p plus two would be excellent 13.

So it holds true for that.

What about if a P is 17 then P plus two would be 19? So that is also true.

Now it's not saying that the difference between every pair of prime numbers is two, but it's saying that there are infinitely many prime gaps of two for going on forever.

So if this is true, then there would be that if it is false, then that would mean that at some point along the number line, that stops being prime gaps of two, and it increases to four or six, or who knows what it increases to now, as yet, it's not been proven that this is true, but it has been shown to hold true to numbers that have more than 380,000 decimal digits.

So we can do that using computers and that's, what's been found.

It's also true that they're so common, numbers greater than a hundred.

You don't have to look very far numbers greater than the thousands.

You don't have to look very far, not going to tell you those cause that's going to form part of your independent task.

So pause the video now navigate to the independent task.

And when you're ready to go through some answers, resume the video.

Good luck.

How did you do on that? Did you find the first pair of twin primes greater than a hundred? You really don't have to look very far for these 101, 103.

So the first two prime numbers greater than a hundred that have a gap of two, the first pair of twin primes greater than a thousand, 1,019 and 1,021, the first pair of twin Primes greater than 1 million, is there 1,000,037 and 1,000,039.

So you see, we really don't have to look very far.

That means that they're occurring very, very often greater than a billion, 1,000,000,007 and 1,000,000,009 and greater than 1000 billion.

We get 1000,000,000,061 and 1000,000,000,063, which is incredible.

It's mind blowing, isn't it? That we can go all the way up to these huge numbers and still find prime gaps too really easily.

So looking at the explore task now, then Alphonse de Polignac in 1849, wrote this conjecture, "For any positive, even number n there are infinitely many prime gaps of size n" So this is the first time we actually see some sort of variation of the print of the twin prime conjecture written down.

But it's thought to date back to maybe the ancient Greeks.

So I want you now to think about how is the Polignac's conjecture different to the original twin prime conjecture that we've talked about.

And can you find examples of prime pairs where n is less than or equal to 20, pause the video now and have a go at this.

Excellent.

So let's talk about it together then.

So how is it different to the original twin prime conjecture? The original twin prime conjecture said that it had infinitely many prime gaps of two, and the Polignac's conjecture says that we have infinitely many prime gaps of size n where n is an even number.

So that means infinitely many prime gaps of size four, six, eight, 20,200, 2000 and so on.

So that's how it differs.

And can you find examples of prime pairs where n is less than or equal to 20? Well, I'm going to leave that with you.

Maybe you can find prime gaps where n is an even larger even number.

So I will leave that with you, to think about, interestingly, quite recently, there's been a piece of research done in 2013 that did manage to prove something about prime numbers.

And that proved that there are infinitely many prime gaps of an even number anywhere between two and 70 million.

So it's not proven that there are infinitely many prime gaps of two or infinitely many prime gaps of 70 million, but have some even number between two and 70 million, it has been proven that those occur infinitely often.

So this was a big breakthrough because it was the first time anything was proven to do with the twin prime conjecture.

I really hope that you've enjoyed learning about the twin prime conjecture.

I've really enjoyed teaching it to you and please, please go out there and research it some more.

If it's of interest, we need more.

Great mathematicians.

Thank you again for watching today's lesson.

Don't forget to go and take the end of lesson quiz and hopefully I'll see you again soon.

Bye.