Large Gaps between Primes - Numberphile

Key Takeaways

It's a fairly natural question if you're looking at the sequence of prime numbers, say P1 is 2, P2 is 3, and so on, how big do these gaps get? And we know that actually the gaps can get pretty big. So there's a very high school argument that if you take the numbers 1 * 2 * 3 up to say * 100, and then you look at this number + 2, this is going to be a multiple of 2 because 2 is a multiple of 2, and this first big number of product of numbers between 1 and 100 is multiple of 2. Another way that mathematicians might write this is 100 {exclamation mark} + 2. So this is multiply all numbers up to 100 together and then add 2. But then exactly the same argument means that 100! + 3 is going to be a multiple of 3. And so in particular 100! + 2 can't be prime because it's a multiple of 2 and it's bigger than 2. 100! + 3 can't be prime because it's a multiple of 3 and bigger than 3. And similarly, you can go all the way on to 100! + 100, and this can't be a prime because it's a multiple of 100 and bigger than 100. And so this is 99 consecutive numbers, none of which can be prime numbers, and so you know that there has to be this gap between prime numbers of size at least 100. But obviously, you can replace 100 here with any number you like, and that shows that there are arbitrarily large gaps between prime numbers. It feels to me you've already It feels to me you've done a proof there. You've proven that there are Right. So this is completely a rigorous mathematical proof that whatever number you like, take it a million, take a billion, there exists a prime gap which is bigger than a billion, or bigger than a million, or bigger than a billion billion billion, or bigger than the number of atoms in the universe. And so, there these really huge gaps between prime numbers, but the prime numbers where these gaps occur are maybe really huge themselves. And then there's this question as to, okay, if I have a prime number of about a million did digits, how big can the gaps between prime numbers be? For example, if you're trying to find a prime number, let's say I challenge you to find find a prime number that's of size about a million. One way you might try and go about it is take a million, try and test if it's prime. If it's not prime, and a million's not prime, test a million and one, test a million and two, test a million and three, until you find a prime number. And this is actually, we believe, a pretty good way of finding prime numbers, that it's completely deterministic, you can do it quite quickly on a computer. Maybe it takes you a while by hand when they up in the millions, but a computer can do this very quickly. However, mathematically, our theoretical understanding of this is pretty poor at the moment. And it could be the case that you have to go on ages and ages before you find a prime number if there were these really large gaps between prime numbers. So, you want to make sure that random dirt you threw at the number line didn't happen to land in the middle of an awesome prime gap. Exactly. Yeah, so if there was some awesome prime gap, this would completely screw up your attempt to find a prime number quickly by just take testing each number in turn, and it might take heat death of the universe before your computer prob- actually finds a prime number, which could be a pretty big problem. So, mathematicians are very interested in saying, given primes of about this size, of size about a million or about a billion, how big can the prime gaps be? And in particular, can we do constructions that are maybe better than this factorial plus two, factorial plus three type constructions to try and understand the large gaps between prime numbers. So, I guess this is the large gaps between primes problem when you asking how large can gaps be, but it really became famous because Paul Erdős very much popularized this problem. Paul Erdős was this very charismatic mathematician. He used to put challenge prizes. So, he put numerical bits of money on different problems. So, he would say, "If you can solve this problem that I don't know how to solve, I'll give you $50. Here's a $20 problem. Here's a $100 problem." And he used to do this as a great way of encouraging mathematics amongst the community. The largest amount of money he put on any problem was $10,000, and he put it on precisely this large gaps between primes problem. He didn't want someone to completely solve all we could possibly ask for about large gaps between primes, cuz that would be too ambitious. However, he and a few other mathematicians, notably Robert Rankin and Besicovitch had a method of doing slightly better than this. Multiply all the numbers up to 100, and then add two, add three, add five. But, it was only slightly better, and somehow that's the only way we knew of constructing what were really larger than expected gaps between prime numbers. It was kind of frustrating for mathematicians that we had this one way of constructing large gaps between prime numbers, and we had no other ways that did any better than random. Erdős's problem was just to try and do better than what they'd done. So, independently, there were two different proofs of this Erdős challenge problem using rather different techniques, um but both based on similar ideas to the work that'd been done previously, um on consecutive days. They came out on consecutive days. I worked on this problem, and also there was a collaboration between Terry Tao, Ben Green, Kevin Ford, and Suchak Konyagin, who came up with a different way of getting of solving Erdős's problem and getting larger gaps between prime numbers. So, we knew about each other, but I was the slow one. They got there had one day ahead of me. We have this website where we put up new papers, and yeah, they got in one day ahead of me. I was the second one. So, they were different ideas. So, as I said before, to improve on what Erdos and his collaborators had done, you really needed to get some new arithmetic understanding about the prime numbers. In my work, it turned out that you could use some new arithmetic understanding about the prime numbers that was a generalization of some of the ideas that had been used to work on bounded gaps between primes. So, somehow bizarrely, if you reinterpreted the whole ideas behind small gaps between prime numbers, they could be useful to get a tiny improvement in our knowledge about large gaps between prime numbers if you took a different understanding of the whole thing. Whereas, the other paper used ideas based on Ben Green and Terry Tao's previous work on long arithmetic progressions in the primes. Quite famously, around 2005, Ben Green and Terry Tao proved that you get arbitrarily long arithmetic progressions in the primes. So, if you just take one prime, you can add a constant amount, add a constant amount, add a constant amount, add a constant amount, and you get these strings of prime numbers. And they showed you can, if you choose the starting prime correctly and you choose the difference correctly, you can get arbitrarily long strings of prime numbers. And again, reinterpreted in a rather different way, they were able to use this extra arithmetic information about the primes in the proof of the ideas of large gaps between prime numbers to solve Erdos's challenge problem and do better than what Erdos and his original collaborators had done. Who gets the 10,000 bucks? Uh so, um uh the 10,000 bucks is slightly complicated, I think, cuz Erdos is deceased, but Ron Graham was in charge of it, and I think he's decided to split it between uh the group of us. What's the new state of play? What do we now know about the gaps between So, it's a slightly technical statement, but I'll try and write it out for you. So, if you look at primes less than X, so I'm thinking of X as being some huge number like a billion or the number of atoms in the universe, then there is a gap of size and now we have some big messy expression involving lots of logarithms. So, log X * log of log of X * log of log of log of log of X / log of log of log of X * some small constant C if X is large. And so, this is a slightly complicated and slightly messy mathematical expression for if I'm looking at primes of size X, how big a gap can I actively construct? Um but this is the current quantity. It's some quantity that gets bigger and bigger and bigger as X gets bigger and bigger and bigger. Um and this is the current state of the art. So, if you want to try and beat our work, I guess I think you need to come up with some new arithmetic information and understanding about the primes. So, the challenge is to do better than this function that's growing very slowly in X. But it turns out that that's completely false, that we really don't understand these guys at all. Um we really have the most limited theoretical understanding on them, and despite being the most fundamental objects most of the most simple questions that you could possibly ask about prime numbers have been open for thousands and thousands of years, and the most famous mathematicians have thought