Gaps between Primes - Numberphile

Hello. I was going to ask you to describe this in terms that you would describe it to say your daughter and then I remembered your daughter does economics at Cambridge. So let's not do that. Let's let's describe this as you should describe it to me maybe. Okay. So there's been a very exciting breakthrough in the uh in the field of number theory. You know, it's caused an awful lot of excitement amongst the mathematicians as excited as mathematicians can get. And the the crazy thing about it is that it's come from somebody who's who's pretty much unknown. It's a guy called Yeang Zang, which is a pretty cool name. And um he's he actually he does work at the University of New Hampshire. It's about prime numbers, the things that certainly got me into maths. In fact, he had he really struggled to get an academic job. He uh he worked for a time in Subway. There are some amazing properties of primes and um they've led to lots of conjectures that haven't yet been proven. There's nothing wrong with working in subway, but you know, normally these the these uh these proves these these breakthroughs are sort of achieved by those that are working at Princeton, Harvard, these kinds of, you know, really elite places. And now we've got somebody who who's literally come out of nowhere that no one expected to to produce this kind of result and and has and has uh has done something really impressive that many great minds were unable to do. But one in particular um doesn't involve multiplication of primes. involves sort of additions of primes and it's the fact that uh there seem to be an endless series of primes which differ by two right so we we can the the obvious ones are the low number primes so three and five and five and seven 11 and 13 these two prime numbers are called twin primes okay and are called twins because they differ by this number two and there's a conjecture that goes back hundreds of years which says actually there's an infinite number of these so the highest highest known pair is remarkable, right? 3 trillion756 billion81,695,685 [Music] times 2 to the power of 666,689 + one is the higher of the pairs of primes. And if I take away one, it gives me the lower of the pairs of primes. That's that's epic. It's an epic, you know, just to remind you, the the lower ones that we were describing were three and five and uh five and seven, etc. So to go to be able to do that and show that that's a pair of primes that differ by two is remarkable. So these the ones that differ by two are called are called twin primes. You also get of course ones that differ by four. Okay, these are called cousin primes and there's even those that differ by six and these are called sexy primes. Okay, which I think you've done as well. Why can't you have prime numbers that differ by seven? You can't have prime numbers that differ by seven. Yeah, because one of them will be an even number. Exactly. Really? Well done. Yes. So, so we know that that there definitely are an infinite number of prime numbers. And I can prove that for you if you want. We've done that. You've done that. I thought you had. Okay. So, so you you know that there's an infinite number of prime numbers. What's not people aren't sure about is that there are an infinite number of prime numbers that differ by two, but it's believed to be true. And so the the the goal is to try and show this and it's never been shown. But um what has been shown for the first time is that you can bound uh the the difference between two primes that the and somebody has shown in fact Zang Yeang Jang from the University of New Hampshire has shown that um there is a bound between two primes let's say uh one prime A and another prime B and that bound is that you know it it can be some number n and so n would be two for the case that we're interested in here and that's the ultimate case that people are interested in. But what he's managed to show is there is some number n for which for an infinite number of of primes a and b this is going to be less than or equal to uh 70 million. Okay. So just to be clear two primes can be separated by more than 70 million. Oh yes yes yes yes they can. So what he's shown is that and in fact the conjecture is that um there is every single even number there is an infinite number of primes that can be separated by that amount. So here is the even number is two right. So there's the conjecture is there's an infinite number of primes uh pairs of primes which are separated by two. But there's also a conjecture that there's an infinite number of uh pairs of primes separated by four and an infinite number separated by six and eight and in fact up to infinity. So that all the even numbers the the conjectures are there are an infinite number of primes separated by that amount. So what he's but that no one has been able to show that's true of any number up to now. And what he has demonstrated is there are an infinite number of primes which will be separated by an a number n which he hasn't yet calculated but he knows that it's less than 70 million. There are an infinity of these guys. Oh god. Hello. Hi. I'm in the middle of doing a video. Well, I've got to answer it so it stops ringing. All right. Call you back when we're done. All right. Is that it? No, is there the mathematicians who work on prime numbers will now no doubt be scouring over what he has done and and trying to uh knock this number down. I I mean I was already hearing about uh one of the key people involved a guy called Goldston who's talked about it might be immediately possible to knock this down to about 16. Okay. And that's pretty a lot closer to uh two than 70 million. But of course 70 mill he has a very nice way of describing this value. Maybe 70 million means the prime the primes are not twins but they're certainly siblings. Why is why is it amazing I think is more of the point. Why why is it really incredible? Well there's a sort of nice way to illustrate this. One thing we know is that there that obviously there are infinite number of prime numbers but that the gaps between the prime numbers generically get bigger and bigger and bigger. In fact, you know that the average for the for the first n for prime numbers between zero and n. Okay, the average gap is of order log of n. Okay, it's just it's just sort of it's a function but this is a big number. Okay, is the point. It's not as big as n, but it's it's a big number. Okay, so so okay, so let me illustrate what that means in in practice. So imagine you had a scenario where uh you've got all a world with all the numbers. Okay? And um there's some rule and I'm just going to impose this rule because I'm king of this world that says that you know prime numbers can only fall in love with other prime numbers. Okay. So the idea is that you go on dates with your nearest neighbors. Okay. And you know do you fall in love or not? Okay. So, so the for the prime numbers at the lower end, you know, the number spectrum, they've got it made. You know, three can, you know, gets it on with five, seven's getting on with 11. You know, they don't have to go very far before they find their true love. But when you get up to like say a Google Plex, right? In principle, on average, you expect to go on a of order a Google date before you're likely to find your um you know, your true love because the prime numbers are so far apart at that large end of things. So, it's a pretty loveless place, you know, at that end of things. So, you go to bigger and bigger numbers, you might think there's just no way you're going to find your true love and you you probably wouldn't even bother going out of the house. You just stay in and watch, you know, Jeremy Kyle or something. But you know what what is actually true there what what Zang has shown us is that for some lucky prime numbers at that very high end of things they actually and it's always the case there are some that actually will only have to go on about 70 million dates before they uh they find their true love. So there are always some prime numbers which are relatively close together. 70 million seems such an arbitrary number. Yeah. And it's like it how if it's possible to explain how has that fallen out of this group? Two, three, four, five, six. Okay. So um so so how does how when people do number theory, how do they actually go about doing these proofs? They they tend to use civ theory.