Prime Gaps and Recent Progress

okay so P1 is two P2 is three and so forth okay so we can look at Prime gaps which is just the difference between One Prime and the next okay given One Prime PN how long do you have to advance before you see the next prime okay so the first Prime Gap is one after that they're all even numbers because all primes are odd after two so uh the prime number Prime Gap starts at one and then you have got two two um four and so forth um and so you have the sequence of even numbers and uh you can you can ask two basic questions about this this Prime Gap okay so one is how small can this Prime Gap be in terms of n for large n and how large Okay so almost the most basic questions you can ask okay um and in both of these questions we have a conjecture which looks impossible to solve by our current methods uh but we also have progress and in both cases actually we had progress in the last two years where this nice symmetry actually um and ultimately as it turns out we actually use the same method to to attack both questions which is not which was not obvious actually uh when we started this okay so I'll start with the first question okay so the uh the basic conjecture here is one that most of you probably know is the TRN Prime conjecture that there infinitely many prime Twins and infin infinitely many pairs of primes that are distance two apart so another way of saying that is that the prime Gap is equal to two infinitely often okay um as the primes as you go out into the primes we know the primes do get spar sparer so on the average the prime Gap actually increases uh roughly like the logarithm of the primes actually um but uh but we still expect every so often just by pure chance basically that the prime should the prime Gap should every now and then return back to be as small as it can be which is two remember has to be even okay so this is still open this was uh first explicitly posed by DEC in 1860 something uh and it's still open um and um for a long time it was basically just seen as as completely out of reach um but uh we had this amazing breakthrough byang Jang last year and the precise date is actually May 14th 20 2013 uh who for the first time managed to get a bound which looks like this uh what he showed was that uh the prime Gap is bounded infinitely often and the bound he even got an explicit bound pretty large but explicit okay that um that there are in that there are infinitely many pairs of adjacent primes whose distance between each other is not two well could be two but is that but is certainly bounded by some by some large but but fixed continent so this this is the first bound of this type so so previously the best bound like this um just for comparison uh uh one could get a bound which is basically a square root of logorithm infinitely often uh this is what goldon p and ym uh about 10 years ago okay this this was the first actual bound um so this was an amazing result um you know it's now published in the anal of mathematics uh it's amazing for several reasons uh partly because of yutang's life story you know he was he was a um an assistant professor in the University of New Hampshire who uh actually hadn't published for a while and even was left Academia for a while but still came still was working on this problem and actually solved it and were completely correct proof um so uh it was uh so there's nothing too special about this number 70 million it it's what came out of of of yutang's argument and he wasn't completely careful he his goal was not to get the sharpest possible number just to get S of a nice round number which was large enough for everything that that he wanted to do and so very quickly after his his pre-int was made available people people started noticing oh if I if I if I optimize this line I can I can change 70 million down to 63 million and somebody else said oh I can I can shave it down 58 Million by this extra trick and so just so spontaneously people started um uh going through the papers and and and finding uh know very very briefly holding the world record for the the BR of gaps uh the G the BR of gas primes and So eventually uh myself and and several other people organize this into an online um project which we call a polymath project as a play on Word polymath means someone who is good at many many different many things but here we mean it to mean many mathematicians poly mathematician working together uh each sort of sort of focusing on one aspect of of of J's argument and and some doing some numerics and other people adding some some theoretical advances and so we worked uh for uh online for uh several months uh and we managed to improve his bound I think by July 27th we managed to get the bound down to about 4,000 and then at which point we began to get stuck uh and we started writing up the results um so okay we're all very happy with that um but actually before we had finished writing up uh actually it turns out was almost 160 Pages worth of stuff actually calculations that we had to actually write up um it's just been trimmed down a bit but um but before we finished actually we found um we were surprised that um independently of our work um but probably inspired by by what y Tang did a postto at at Montreal his name was James mayard and the time here was uh November 19th found um a shortcut that simple that went uh that cut out a lot of the difficulty in jang's arent Jang used quite a bit of Firepower uh just give you one example uh there's a very famous result of delene um uh proving the wean hypothesis over finite f uh and it won him the fields medal and the Apper prize and so forth uh and Jang used that um to uh to get this um and we also did as well um but Maya found a shortcut that that that cut around a lot of this and uh so he managed to find a much shorter argument which of of the same okay but he got a much better bound of 600 actually okay um so after that came out you so we my group got very excited and we started talking to Maya and so he decided to joined forces with us and so we we worked a little bit more to combine our two methods and so uh we worked on that for quite a while and we eventually got uh yeah so that the the final thing that we uh that we declared victory at uh was back in April okay so our bound is now 246 and it's possible with lots of computer power to maybe shape another four or six out of this but uh but it it got really complicated and if we thought this is a good place to stop now um this this was a I mean um for the purposes of an online project it was really good to have this number as sort of a score right because you could you could it was a very tangible measure of progress which is something that you don't often get in a math in a mathematics project uh I should say though that this there's nothing that special about these numbers um you know I mean it's it's um what's more interesting ultimately is the techniques that go to proving these things and the various advances in technology that you need to get get from one to the next uh but still it's nice to to have this this this very clean progression here um if uh um but unfortunately you know I mean this looks like it's approaching too but uh slowly but um we we we unfortunately we do know that that these methods cannot um attain the tri time conjecture so um the most um we what we can prove now is that if you assume um so all these results are proven using the methods of C Theory which I'll talk about later um but um and there's a certain conjecture in C the which is kind of like the r hypothesis um something called the hem conjecture I'll explain what it is a little bit later well roughly um but it's it's sort of like a generalized ran hypothesis for C theorists uh and if it is if it is true uh we can we can improve this bound all the way to six that we can get infinitely many many pairs of primes six apart um so pairs of primes that are two apart are called twin primes uh pairs of of primes that are four apart are called cousin primes and pairs of primes that are six apart are called sexy primes uh so this would mean that there's either infam twin primes INF cousant primes or INF sexy primes so cute phrasing it uh in yeah um but unfortunately um we know also that this is the limit of the method that there is no no purely C theoretic method that can that can lower six to anything smaller um so if you want to actually get all the way to two you have to use something different from what anything that we do here um okay so this is this is uh uh these are some of the results we have we have other results as well um so one way to think about these results for example this result here is saying that you can find infinitely many intervals of length 600 which contain two primes um it turns out that that some of the later methods starting with mainard and and going forward can um also uh find um intervals which contain many primes um so so not just so which which means um so rather than than talk about prime gaps you can also talk about you know p n plus 2 minus PN and there there's some Bound for this and there's Bound for PN plus 3 minus PN you can find intervals of bounded length that contain three primes four primes and so forth um and this this this will become somewhat relevant later on uh but there there are other um variants of this and so the methods that been used to prove this uh these Prime gaps results have been also been used to prove many other things in number Theory um there's actually now know this is only like like 12 months ago but there's already like like a dozen papers using variants of what we now call the main ariv um or may not talve by some people okay to um to to prove other things okay so this is the state-ofthe-art for small Prime gaps uh now let May briefly talk about large Prime gaps um okay so large um a large Prime Gap is the same thing as a big long string of consecutive of composite numbers okay if you a whole bunch of composite numbers then the prime before and the prime the first Prime um to the left of that string and the first by that string has got a big gap so uh one of the first things you can prove and this is something some sometimes you even see in high school is that the prime Gap can be infinitely large so so this Gap can be arbitr large and there a very simple proof given any number n you take as factorial and you look at this consecutive string of numbers all the numbers from n factorial plus 2 up to n factorial plus n let's say n is at least two okay and these are all composite numbers because this guy is a multiple of two which which is bigger than two so it's composite this guy's M of three which is bigger than three so it's composite and so forth so youve got this this whole string of of um composite numbers of length n minus one so that means that that the primes to left and right of of this string must have a gap of at least n minus one maybe even n and n can be as big as you want so the prime gaps can be as big as you want now we can do better than that by using some slightly more advanced technology um so um okay so one of the first basic themes that we can prove about the primes maybe the most important thing about the primes really it's the prime number theorem proven in 18 uh 990 something I should know this okay but uh okay I think 1896 or 1897 um okay so uh the prime number theorem tells you roughly how many primes there are up to any given uh level so tells you that the number of primes up to a large number X is roughly X on log X and this is a natural log of x okay and roughly what you mean is that you actually have to multiply by a little error which goes to zero in in relative sense so the um that the ratio between the number of primes up to x and x log X goes to goes to zero goes to one as X goes Infinity okay so for a large number X like a trillion the number of primes up to a trillion is roughly a trillion over natural log of a trillion so that's how many primes there are in this interval and so just from the original principle okay uh what this tells you uh is that um among the primes so among the primes up to X there is at least one prime Gap um PN which is bigger than log X maybe a little error okay um or another way of saying that is that uh is an equivalent way is that the prime Gap can be as big as uh small factor Times log PN the prime Gap can be as big as the logarithm of of of the of the prime of say the first Prime infinitely often okay so this is a an easy consequence of the prime number theorem and the um and the pigal principle okay so um but okay but can you do any better okay can you make the prime can can the prime gaps be any bigger than this so um we don't actually know so much about how big the prime can be um I think the best unconditional bound we know is that the prime gaps are always at most a constant times PN to the I think 0.55 um this is so they can't be incredibly huge like like they can't be bigger than than than um uh 0.5 um basically almost the square root of PN this is a theorem I think of of hoo um no bam and pins this one um okay but um if you assume if you assume the r hypothesis uh you can get a slight Improvement uh we know that this that you can improve this to PN to to the one half and Times log PN the 055 yes yes it's it's a uh just a numerical constant coming from optimizing all kinds of other things um yeah but um if you the we hypothesis we can bound the prim up to the square root of of PN basically but that's still quite far from this pigeon hole um and this is not expected to be the truth um but this is just the best we can do even assuming this very heavy heavy machinery here which is the which is the r hypothesis um so the um the conjecture that everyone believes is J to Kramer so uh he used uh he introduced a random model of the primes uh so so he he modeled the primes so the primes up to X are just some set of numbers up to X okay and we know how many numbers there are but uh of course knowing the the number the cardinality of the set doesn't tell you everything about the set it could it doesn't tell his distribution uh so so krer had the idea well let's pretend that um this set we know is cardinality let's pretend that it looks random so he he modeled the primes by a random subset of numbers up to X of the same cardinality of cardinality approximately X of log X so in a way it's sort of giving up on the question of of what patterns there are on the primes let say I don't understand the set of primes let's just believe it's it is a random set of course the primes themselves are not actually random you you don't have to roll any die or flip any coins to to create the the P numbers but um the modern way of thinking of of Kramer's model is to is to say that the primes behave sto randomly that they behave like a random set even though they're not ex exactly random um now the reason you want to do this is because uh we have this uh probability theories are very good at Computing what happens to random sets and and we can compute all kinds of statistics of random sets uh so for example if you take a random set of numbers up to X of this cardinality you will find a lot of twins a lot of pairs of numbers two apart um quite a lot actually and so you uh this this gives you some evidence some htic evidence that t Prime conjecture is true um if you take this random subset uh and you look at Prime gaps between now these sort of fake primes these sort of random um model of the Prime what you find is that the the prime gaps and now the this this random variable they'll be distributed like a proon random variable uh whose um um uh sorry an exponential random variable um yeah the set looks like a Plus on process but the the gaps have this exponential variable of of mean log X uh and so you you you can get gaps larger than log X but they get exponentially rarer um you know as as you make the Gap bigger and bigger and um by Computing the random um by Computing the random um uh model krer conjectured that the correct size of the Gap should be log PN squ okay that if you make a sufficiently for a small constant C for espicially small C okay um that if you make this constant C small enough you should have gaps that are as large as the square of logarithm infinitely often but then he also conjectured that if you make the constant big um this should only happen finally often um okay he maybe he didn't quite conjecture that but this this is this is what the uh uh this is what the same model would would would predict for you okay that's that the largest gap should be roughly of the order of log squared okay you you get an extra log over what the the basic equal principle gives you okay but this this conjecture looks really far from reach um so I mean the the Trin Prime conjecture I I can believe that we just one or one or two clever ideas away from that you solving this um I mean we know C Theory by itself can't do it but maybe we just need one or two more clever ideas and we could maybe do this conjecture um this I don't think we're anywhere close to to um to solving um one of the reasons is that twin primes are still fairly numerous um at least we believe them to be fairly numerous among all the Primes so it said that the number of primes up to x is by X over log X um the random humanistics that come from this crma model suggest the number of twin primes up to X is about X log squ X U which is not which is still quite a lot of quite a large number there's a lot of twin primes and you just need to find one twin Prime in every say inter between X and 2x for large x um it it doesn't look uh completely impossible but with this uh with the large Prime gaps the larger you make your Gap the fewer and fewer um Prime gaps you have with um with that large gap well Le at least that's what we believe and and the density decrease exponentially so for example um we um yeah there should only be like one prime Gap this this huge in any interval between X and 2x or something like that um and so these are very hard to find uh these these huge large these huge gaps so I don't think we're anywhere near as close to this conjecture as we are to the CL conjecture although this one is also we are still quite far from that too unfortunately but we have some progress okay so it turns out between log log X and log and log squ X it turns out to be a surprising amount of of intermediate room um at least to a number theorist so those are sequence of results and it's it's a um puty well the list is somewhat amusing um so the first nontrivial result was by uh I can't pronounce the name polish mathematician West zinthus in 1931 who managed to beat the positional principle who's the first person to beat the positional principle and he showed that you can get Prime gaps that are as large as log PN times a correction term and his correction term turned out to be uh log log log PN except he could couldn't quite get that so he had to divide by log log log log PN okay uh there is a joke not a very good joke but yeah what do you what did the drowning number theor say and the the answer is log log log log log log log um but anyway okay so he was the first to get um uh to get to get some non true your improvement uh this was quickly improved well not so quickly but by by erish in 1935 I thought this is true infinitely often of course okay um and the same result uh but now slightly less embarrassing logs okay two logs on the top and uh three logs on the bottom squared okay um and then Ranken 938 uh managed to to to uh to improve this just a little bit more he could get um um Prime G as large as a constant times log PN okay again two logs over three log squar same bound okay and then he um was able to add an extra fourth log up here okay and C was I think one3 when in his first paper okay so you know okay so there there was there was some progress here um but amazingly what happened afterwards was that this uh uh people got stuck at at this at this Bound for for for uh for over 70 years um well so what had so over the next few decades people were able to improve c a little bit but they could not get um get past this this funny natural expression of of of of a combination of logs um so yes so C started steady improving there were results of uh um Sean hauger and and then ranking again May and pomerans and then finally pins until until this year um the most recent uh improve result on this was was this the same result just C was a bit bigger C was actually two * e to oiless constant which is a number about 3.56 something okay but other than that there was no further um there was no gain to this in fact this was so um stubborn and so mysterious that um erish who was fond of of of of giving out cash prizes for his problems actually offered $5,000 to anyone who could actually improve um to show that this constant could be taken arbitrarily large which is the same thing as as basically adding some some extra log or something to to this uh to this to this expression here um so this this is uh this is where things were stuck at for some some time um but then there's a a breakthrough this year and the funny thing about this subject sometimes is that um sometimes the time is just right for for uh for a breakthrough you know so mayard for example uh uh was was working independently of Jang on on on Prime gaps and the people were stuck for 10 years and then there were these two results which are pretty much independent of each other May was already working on this problem before Jang um and so we now have these two new methods of proving B gaps and the same thing happened here with large gaps so um uh so very recently there was a result of Kevin Ford Ben green green C Kagen and myself and uh this was uh 20th of August um okay so what we proved is that you can take C arbitrarily large okay so so this is true okay that you can get a tiny improvement over this um and amazingly uh the next day um this has happened to be before not not actually once it actually has happened 24 hours apart uh but sometimes I think just the the ideas in the air um so James mayard had released a proof from the next day proving the same result um by a different method and yeah so it was truly independent and actually the amusing thing is that he used results that we used to prove small Prime gaps to also to prove large Prime gaps which is which is an ni symmetry um so since then uh We've joined forces and um we can now um um actually push this a little bit further so so um as of I think last week uh we can so all together uh well if you put everything together what we can do is that we can get rid of one of the log log logs on the bottom okay yeah so you can see we're not really getting all that close to clus conjecture of log squar um this this is these are still quite small improvements um and and there is a serious limitation to our methods we're not going to get anywhere close to to to CR this actually we believe is the limit the new limit of of our methods you will need a new idea to get to get Beyond this particular um thing um so yeah but but still it is nice progress and it uses um some of the theory we had before what would to oh um this year year yeah yes yes Ben actually announced the ICM in Korea uh okay okay so uh these are the results um so let's talk about some of how we prove things let's go back to small gaps okay so um okay so we have the the pigeon principle gives you something okay so the same pigeon argument that that gives you large Prime gaps also gives you gives you small Prime gaps so again the prim number the that for any large x the number of primes less than X or let's say between X and 2x okay if you take an interval between X and 2x the number of primes in this in this interval is is basically X on log X um and so again from the original principle um okay this tells you that there exists a prime Gap in the inter between X and 2x of of length Okay of the log X or less okay so just from Counting because the total the sum of all the gaps adds up to X and there's about X over log X of them so just from the pral principle there has to be one slightly one gap which is less than equal to the average spacing okay um so but but we want to do better than this we want we want a bounded um Prime Gap not not a gap that goes that goes logarithmically so um all right so how can you proceed well I can rewrite this argument so this is a very simple argument pigal argument but I can rephrase it in problemistic terms so uh problemistic reformulation okay so um so what I'm going to do is that I'm just going to find a prime um um I will I I will um look for a prime randomly so what I will do is that I will um Pick N pick a natural number and uniformly random between X and 2x okay so just pick some some number uniformly random from X and 2x it's a random number it may be prime it may not be prime it will probably not be prime but there is some chance that that it is prime you can confuse the number the probability that you're Prime and um from the prime number theorem um there are about X numbers between X and 2x if this if x over logx is prime the the probability that that n is prime is roughly about about one on log X okay so if you pick a number at random you've got about One log log X chance of of it being Prime okay that's not very big but it's something okay um similarly um if you look at the probability n plus1 is prime if you take a random number X and 2x and you add one to it that's basically another random number between X and 2x not quite but essentially it is and so again the prime number the also tells you that the probability that this is prime is also about one in log X and so on and so forth okay so you can start you can you can take um you can shift your random number by various shifts 0 one two up to some shift say Capital H and H is going to be of size about log X okay okay so you can take your your random number and shift it um H times and each time you shift it you've got another event that this shifted number might be Prime and there's some probability about one in log X that this is prime now if you pick H to be of size just a little bit bigger than log X you can make all these probabilties sum to be bigger than one so you add up all these probabilities individually they're small but if you keep adding them together they eventually get big so for for about the size you can make this the total sum of this guy okay you can make the total probability of all these events exceed one once you you make Capital H H big enough and about log X okay and the moment you do that the probabilistic positional principle tells you that if you have any set of events and the prob the total probility add up to to um to more than one then somewhere with positive probability two of these events happen together okay they can't be destroyed so somewhere for some n there must be two shifts n plus H1 n plus H2 where H1 H2 are of the size where they're both Prime and that gives you a prime gap of size less than log X okay so this is a very complicated sounding reformulation of the opal principal argument uh but this is a good reformulation and the reason that is good is because it generalizes okay so um this argument is very slick but it's sort of hard to see how to generalize it um this argument um okay so so first of all um rather than than taking um consecutive shifts you could you could be maybe slightly clever and take targeted shifts you know may shift H1 H2 you can you can shift your random number by by some well chosen shifts and as long as you can make the total okay and uh okay and this this probability is equal to something this prob something else as long as you can make the total probability of these guys being Prime add up to more than one you can and if these hes are not are not too far apart from each other you can create a fairly small Prime Gap from um from so you can try to play with with different hes um although if you pick n uniformly at random uh terms no matter how you shift you always get one of a logx and so you can't do much better than this log X bound that I already said but the um the but the more relevant Thing Once you once you phas things this way uh is that once you face things publicistic you begin to see that there's really no reason that you have to pick n uniformly um that uh that that maybe if you pick n not uniformly at random but some other well chosen probity distribution maybe um you could pick a better distribution from num between X and 2x such that you've got a better chance of of making these um these guys Prime and so that you can increase um each individual probability and then maybe you need fewer of these terms is some to be bigger than one and maybe much less than log X okay so you know for example if you want n and n plus 2 and N plus 4 to be Prime and so forth um picking n uniformly at random and hoping that you get a prime is kind of silly you should at the bare minimum only search um the odd numbers if you want n or n plus 2 N plus 4 to be prime okay um that already doubles basically all the probabilities here okay so that that already gives you um gives you a win the price you pay though is that the hes have to become further apart you can't take consecutive numbers you have to take consecutive even numbers um but that already gives you some some uh some idea that you can play some games here that rather than work of a uniform distribution maybe work of a uniform odd number or maybe a uniform odd number which is also co- Prime to three or something like that and so you can start um so um so phrasing things in a in a non- probalistic manner um basically uh you have a freedom to choose a probity distribution now so so now the the new goal is to find a prob distribution or equivalently basically a series of Weights a sequence of positive numbers for n between X and 2x such that uh if you um so use these weights to uh to create a prob prob distribution by normalizing by the sum okay so um so you you you some these up you get some number and uh then you you sum up um all the ends for which say n plus H1 is prime okay so this ratio the the probalistic interpretation of this ratio is this is the probability that if you choose n using this weighted uh this particular set of Weights as a distribution that that that you get a prime when shift by H1 and then you shift by so if you shift you do all these shifts and if if you can addable these shifts and you can get this this thing bigger than one um this this implies that for some n at least two of the N plus h Prime okay which and if the H is the hi you the shifts are fairly close together you can get a fairly narrow Prime Gap in this manner um also you know if you can make this this ratio so big that is bigger than two then not only can you get two of these guys Prime you can get then get three of these guys Prime and so on and so forth so that sort of brings us to other things I mentioned before okay so um so basically all you need to do is that you need to find a clever choice of weight um which on the one hand you want the total sum of the total mass should not be too big you want this denominator to be small but you want this these numerators to be big uh you want uh um you so you want to find B what's called a Civ which which U which catches a lot of primes that um that that relative to Total size the portion of of of of the sum coming from primes is is relatively big much bigger than one of a log X okay so one just has to pick um this this weight properly um so um so the game then is is to find a very clever choice of of weight which does this so you know um so you could try the the uniform weight but that that that doesn't work very well um you could you could be overly clever and and um for example if you wanted to maximize a public that say n is Prime and N plus 2 is prime you could pick n not to be uniform uniform random you could pick n to be a uniform random among the twin primes in from X to 2x so from X to 2x you pick all the twin primes um and then of course you're guaranteed if you do that you guaranteed the N is that probably n is prime is one n plus two is prime is one 1 plus one is bigger than one and you win now why why doesn't that solve the problem um the problem is that uh that corresponds to to to to selecting new event to be one when N is a twin Prime and zero otherwise um and there the problem is that you can't make the you can't prove the denominator is non zero unless that's a hardest original problem um so if you want to make both of these probabilities exactly one uh we can't do that but we don't need to make to get anywhere near that far okay we um you could you could you know if for example you could find a weight where the probility N is prime is at least 6 and plus two is at least 6 that's already good enough um and so you can you can hope to find um um smarter choices of um of this weight which um uh which you can actually for which you can actually compute these sums without having to result to unproven conjecture and then hopefully you can make these numbers big enough that they sum to bigger than than one and then and then you you can declare Victory um so this is this is ultimately what everyone did so this strategy was introduced by goldon Prince in yum about 10 years ago um and that's how they got their first partial results on on on Prime gaps um now it turns out nowadays it turns out that um yeah so the question is is what weights do do you pick um and so on the one hand you want the weights to always be positive because otherwise um you don't get a proba distribution on the other hand you want actually the weights have to be simple enough that you can actually compute these sums um and it turned out after a lot of Tri and error that the the correct choice of Weights that the best choice of Weights that that we have currently come from What's called the silberg C CH by Atley silberg and the 7s I think um and so uh his weights um yeah so if if you want n plus H1 and N plus if you want to make a Civ such that all these guys have a high chance of being Prime um the soic Civ what what uh is uh what you do is that you take you take all you take your all these numbers that you want to be Prime and you multiply them all together and you get one big number and you look at all the divisors of this big number um and then you take some sum uh you take some sum of this um of this um um uh you over you sum some coefficients based on the sum and these are positive or negative coefficients and you take the sum and then you square it okay but you don't take all you don't take all factors you take all factors up up to a certain level so there there's some parameter r that you pick so you take all the factors of this big number up to some level R you you choose some weights and and so you get to pick what weights what weights you have here um and then you square this okay so the reason you square this is to guarantee it's positive so so these guys can be negative but the but the square is always non negative so this is always non- negative um and this turns out to be a fairly tractable expression to to to compute you know so um for example if you want to compute um U of n okay uh if you um if you unpack this this is equal to uh if you square this out this is equal to like the sum of all pairs D1 D2 up to R then there's some weights another weight and then you're summing Over numers N between X and 2x uh one subject to the constraint that D1 and D2 both divide n plus H1 n plus H2 so this is just expanding things out okay and the um the point is that this is something which is actually fairly easy to compute so if you just summing all the numbers from X to 2x this is basically just X um here you are summing all the numbers um n such that this product is divisible by by D1 and D2 so in other words it's divisible by the greatest the least common multiple of D1 and D2 now um if D1 D2 are bounded by R this number is bounded by R squ so as long as you pick R small enough like R less than say root X if you pick R small enough the modulus here is less than x and so this constraint is is a periodic constraint whose period is less than the interval in length of the interval here that you're summing over and that turns out to make this the sum very easy to compute uh and then and then what you get is you get some quadratic form in these coefficients okay and quadratic forms are something that that we don't scare us too much um if you instead of summing all of these guys you're summing um for example um this sum weighted by the the fact that this guy's Prime then you you get a very similar you get a very similar sum over here but rather than summing one you're summing primes uh the indicative function of the primes and so um what this sum is is basic basically roughly speaking this is counting how many primes there are in in arthic progression okay and there are lots and lots of theorems about what primes do in Artic progressions that that you can use some an analytic number Theory so that's so the the number theoretic input is then um to to control this use facts about about the number of primes in arithmetic progressions um so you may know for example dur theorem that if you have any residue class a mod Q where a is co-prime to q that this residue class always get INF minum primes um so that fact by itself is too weak to actually say much here but there are stronger versions of D theorem that that can go into here for example if you know gr you can plun it in here um but even without gr you can there are various ways you can you can compute this this expression and so you get another quadratic form so this this this funny ratio here what it is with the sober because a ratio of two quadratic forms with with with coefficients that you can pick and we know how to optimize quadratic forms right then you can do some linear algebra and you um and um you can start um you can start doing this and so um it turns out that if if you if you plug if you run this machine using um uh using the best theorems about primes andic progressions that were already available before yeang jang's work you you just barely fail you can get this ratio as close to one as you wish but you can't quite make it equal to one which which is very annoying uh so um go and P were not able using this method they so they introduced this method they were not able to get B gas but they they got really really close and the problem was that they didn't have a strong enough theorem about primes in arthic progressions um and jang's big breakthrough was that he found a new theorem about primes and E progressions that was not available before um uh oh okay maybe I I won't actually say exactly what it is um but um if you want to count the number of primes up to X in an arthic progression like a mod Q um this is this turns out to be a fairly easy task if Q is small but when Q gets bigger and bigger um the progression is sparer and sparer it's harder and harder to count the primes um and the standard tool in C theory is something called the bomber vinag theorem which allows you to to count primes in Artic progressions as long as this the space in Q is not too big basically as long as Q is less than x 12 um as long as you don't make the spacing any bigger than than square root of x we have a a very good theorem that tells you what the primes do in most of these arthic progressions um and and what Jang did was was was to be able to stretch this this suum a little bit and also say just a little bit about um primes in arthic progressions which were spacing just a little bit bigger than X2 and but it was a tiny amount I think the amount he goes like one over 568 or something um but that was just enough to tip the balance to make to make this expression just barely bigger than one if K was big enough I think K be something like three million um and this is what how he got his his his 70 million bound um initially uh and then subsequently we were able to push this enlarge this Con further and and optimize other parts of the argument okay so um and J worked quite hard to get that that time this this little epon actually he use a lot of Fury analysis uh a lot of um this work of Del on exponential sums and an incredible number of applications that Kashi Schwarz inequality um but um anyway he was able to to to get that um so mayard later on found that you didn't actually need so he spent a lot of effort to to get to get this Improvement um but actually even without that Improvement using just the existing primum in arthic progressions um Maya was able to to get the same sort of result um and so his observation was actually very simple uh which is that this is this was not the um the U the only C you could you could pick so the standard C until then had been to multiply all the numbers together here and take the devices of this product and create a Civ um but then mayard and actually I realized this independent at the same time it's another instance where uh uh have simultaneous Discovery but um but what you realize is that there's there's a more General C you can pick here so rather than look at the factors of a single of the single product you can just sum of us the separate factors which in retrospect is a very simple idea we should have realized this earlier in fact it had been realized earlier this this idea um but the reason why it was not studied further I think was unfortunately the first paper um to analyze the Civ actually had a serious error in it um and so it was somehow discredited and people didn't pay attention to it until 10 years later um yeah so rather than than look at the factors of this big product uh you split up the factors into into into K pieces um so now you have this multi-dimensional sort of Weights here um but then you can actually turns out you you can run the same game and um you you have another quadratic optimization problem uh and it now in m in many dimensions rather than one dimension um it turns out that it reduces to a certain calculus variations problem that you have some multi-dimensional integral of a certain test function quadratic form of a test function and in both the numerator and denominator and you want to to maximize the ratio um and turns out that um that if you choose the right test function you can actually make this ratio much bigger than than uh than in the one dimensional Cs and this is how he was able to to to get his results where in fact he could make this ratio as big as he wanted he he can get get many primes in B inter not just one prime okay and then afterwards the polym project improv us further uh we we uh we got some experts in in quadratic optimization on board and they they they they started optimizing these huge matrices and uh yeah this is how we we got our B down a little bit further and we made some other improvements too okay so that's all I want to say about small Prime gaps so let me talk now about large Prime gaps so um as I said before there is a simple construction of large Prime gaps so lot finding large Prime gaps is basically the same thing as finding large consecutive strings of of of composite numbers okay and the simple construction is just take n factorial and then you shift it by the numbers from 2 through to n and this already gives you um a large Prime Gap um if you f around a Sterlings formula uh this construction does give you a prime Gap and it tells you that the the side the Gap here is about log PN over log log PN infinitely often this is what comes out Sterling's formula uh so that's not so great this is actually worse than what you get from the prime number theorem prime number the the P already gives you a log PN and you actually want one log log behind if you use this simple construction okay but you can do you can be a bit smarter than this infactor is a bit wasteful um there's a smaller version of INF factorial called n this called the prim orial the the prim orial of so the factorial of n is all the product of all the numbers up to n um natural numbers the prime orial of n is the product of all the Primes of to n so you just you only multiply the prime numbers forget the composite numbers um and the same construction you take the primordial of N and you add two to it add three to it all they have to add in to it you still get all these numbers are composite by exactly the same argument and this this Prim is a bit smaller um so n factorial is bit like n n this is more like exponential of v um and so if you use the primordial which is the smarter thing to do uh you you can recover the ponal principal bound okay so you you're at least not losing anything uh but you haven't gained anything yet either sh sorry what the shs in oh yeah you you use all the numbers here so um yeah you need a consecutive string of composite numbers to get a large Prime Gap if even miss one then um um then you um yeah okay so um so why did this work okay so you can generalize this um so another way to think about this construction is that what we're doing is that we if you look at this interval this block of consecutive numbers from 2 to n um you can cover this interval by residue classes like if you take the mulus of two okay so that crosses out some of these numbers you take the mulus of three so those are basically Theos and if you take the mpol of all the Primes up to up to n every um um each of these resue classes between them they cover this this interval this interval is completely sied out it's completely covered by these rest two classes every number from two to n is divisible by One Prime at least between two and n and it's because of that that that's what makes this um this whole string consist entirely composite numbers because everybody here divisible by some prime you shift it by the primordial which is divisible by all the Primes up to n and so everybody here is divisible by one prime up to n now if you're facing this way what you realize is that uh um more generally if there's any interval you take any interval and if you can cover this interval maybe not by um um the the zero resue classes but if you can find some congruence class mod 2 um maybe not call it a I know it's called C some Cong class Mar 3 and so forth if you can find some congruence class mod um mod two mod 3 mod 5 which together cover every single interval here then um you can shift it um by n factorial maybe not by n factorial but by some other number okay um you can you can use the Chinese remainder theorem to to find some number Y which is I think has to be minus C2 mod 2 minus C3 mod 3 and so forth if you can find so by the Chinese remainder theorem because these all different primes you can find some number between one and the primordial which mod P is minus CP and then if you shift this interal by that Y what you'll find is that is that every number in here is divisible by one of the primes up to n and you would still get a string of consecutive numbers so so really what's going on in this construction is the fact yeah so um is the fact that that this interval between two to n can be covered by resue classes from from of of primes up to up to n and so if you can find a longer um um a longer interval which can also be covered by primes up to n then you can get then you can start beating this result okay so now the question becomes another C thetic question what is the most efficient way if you have a long interval of of numbers what's the most efficient way to to take AR class mod two and model three so you're allow to translate each each one okay but you want to to slide them around in such a way that they cover everything okay so this becomes more of a commentarial problem so it turns out that if you use all the Primes up to n um so the way we prove out our theorems is that so using the prims up to n we we can cover all the numbers from two to n but actually now we can cover all the numbers from from um we can cover a slightly longer interval um so I think the uh the record currently um all right well it's not so important exactly what it is but but uh but we you can cover an interval slightly larger than n using only only only um only primes up to n and so so how do you do this um so um the way you proceed here is that you you sort of um you modify the C osines you you um you use the the the zero congruence class for most primes say all primes up to say n / two okay if you if you C out all the mels of of of all the Primes up to n/ two that already cuts out a lot of numbers um any number that has any prime factor less than two will will be killed uh the only thing that survives or first of all one will survive um that's not very interesting but um but basically the only survivors so the um survivors are mostly just the primes between I think n/2 and this big number let's go y okay so um the primes are basically the only numbers that don't have small prime factors okay um so if you s out all these numbers uh all that's left are the primes and we still have all the Primes so but we still have some primes left over we we we still have some residue classes between n over2 and n to play with to to to sort of kill off any F survivors um as it turns out strictly speaking we don't quite do this um um we s out this for most primes P um and then we we we randomize it some of the pr it's actually better to to pick random R classes rather than than zero uh so there's some there's some optimization here which I I will L gloss over but roughly speaking um using using some of of of of these resue classes you can you can get you can reduce just primes in here and then the question then becomes uh can you cover the remaining surviving set of primes efficiently by congruence classes um coming from from from these primes here um and and so what the problem boils down to is uh can you for given Prime P between say n n/2 and n can you cover the primes okay can you find um a congruence class CP P which catches a lot of primes a lot of times in this interval from Z to one okay but H istically if every prime or most primes p in this in this range for each prime P you can find some congruence class which Each of which captures say hits about 10 Primes from one to Y then um if there are like interal login primes here then between them all these kind should be able to hit maybe 10 10 and Logan um primes out so you and so the more primes that each that that that you can hit with each congruence class um the the more efficient you can make the siing process so this becomes the basic question so um when Ben green Kevin Ford C Kagen and I got interested in this so um this looked very similar to something that we had worked on 10 years ago Ben green and I um so 10 years ago we had studied arthic progressions in the primes and when we we in particular we were able to find nice long arithmetic progressions of of any length you wanted where everybody is prime okay so uh or in other words we could find a congruence class mod R which hit a lot of primes um now for in our first paper this R was just a number it wasn't Prime we had we had a subsequent paper in which we could you could you could basically Force R to be basically a prime or maybe a small model of a prime um so we used um our Machinery from several years ago on Counting things like progressions and primes to be able to to um uh to answer this question but it was quite complicated because we this these results were themselves rather non-trivial um but what James mayai did he seems to have a knack for always finding the simplest way to solve a problem um is um um another way of thinking about this question you know so what you're doing is that you you're taking CP you know you're taking a congruence class which is like a big athic progression and so uh what Ben and I would would doing is that we were spending a lot of effort to make all these numbers prime um but actually for the purposes of the saving um you don't need to make all these numbers prime in order to catch a lot of prime you just need to have many of these numbers prime and you know James from his work on small Gast region primes had had already found a way to find to produce lots and lots of tupos like this where not everybody is prime but you know maybe say 10 of these guys are prime okay so he could find many patterns where where large numbers of these numbers are Prime and and uh but but also but not all of them um and this so basically if you just call this n and you call this H1 and H2 and so forth it's exactly the same problem um that um the the problem of catching um lots of primes inside a congruence class is a special case of of of finding a Civ such that n plus H1 n plus H2 n plus H3 these are all have a large highing Prime so rather unexpectedly the same basic method which underlies the the recent progress on small gas primes has now been used to also create large caspan tries um so yeah that that is a very nice symmetry yeah so we're still seeing what we can do further uh with with these methods although I think for the purpose of small gaps and large gaps we've pretty much reached the limit of what we can do with these ideas but there are other problems in number Theory which hopefully we can do something about okay thank you very much

okay so P1 is two P2 is three and so forth okay so we can look at Prime gaps which is just the difference between One Prime and the next okay given One Prime PN how long do you have to advance before you see the next prime okay so the first Prime Gap is one after that they&#39;re all even numbers because all primes are odd after two so uh the prime number Prime Gap starts at one and then you have got two two um four and so forth um and so you have the sequence of even numbers and uh you can you can ask two basic questions about this this Prime Gap okay so one is how small can this Prime Gap be in terms of n for large n and how large Okay so almost the most basic questions you can ask okay um and in both of these questions we have a conjecture which looks impossible to solve by our current methods uh but we also have progress and in both cases actually we had progress in the last two years where this nice symmetry actually um and ultimately as it turns out we actually use the same method to to attack both questions which is not which was not obvious actually uh when we started this okay so I&#39;ll start with the first question okay so the uh the basic conjecture here is one that most of you probably know is the TRN Prime conjecture that there infinitely many prime Twins and infin infinitely many pairs of primes that are distance two apart so another way of saying that is that the prime Gap is equal to two infinitely often okay um as the primes as you go out into the primes we know the primes do get spar sparer so on the average the prime Gap actually increases uh roughly like the logarithm of the primes actually um but uh but we still expect every so often just by pure chance basically that the prime should the prime Gap should every now and then return back to be as small as it can be which is two remember has to be even okay so this is still open this was uh first explicitly posed by DEC in 1860 something uh and it&#39;s still open um and um for a long time it was basically just seen as as completely out of reach um but uh we had this amazing breakthrough byang Jang last year and the precise date is actually May 14th 20 2013 uh who for the first time managed to get a bound which looks like this uh what he showed was that uh the prime Gap is bounded infinitely often and the bound he even got an explicit bound pretty large but explicit okay that um that there are in that there are infinitely many pairs of adjacent primes whose distance between each other is not two well could be two but is that but is certainly bounded by some by some large but but fixed continent so this this is the first bound of this type so so previously the best bound like this um just for comparison uh uh one could get a bound which is basically a square root of logorithm infinitely often uh this is what goldon p and ym uh about 10 years ago okay this this was the first actual bound um so this was an amazing result um you know it&#39;s now published in the anal of mathematics uh it&#39;s amazing for several reasons uh partly because of yutang&#39;s life story you know he was he was a um an assistant professor in the University of New Hampshire who uh actually hadn&#39;t published for a while and even was left Academia for a while but still came still was working on this problem and actually solved it and were completely correct proof um so uh it was uh so there&#39;s nothing too special about this number 70 million it it&#39;s what came out of of of yutang&#39;s argument and he wasn&#39;t completely careful he his goal was not to get the sharpest possible number just to get S of a nice round number which was large enough for everything that that he wanted to do and so very quickly after his his pre-int was made available people people started noticing oh if I if I if I optimize this line I can I can change 70 million down to 63 million and somebody else said oh I can I can shave it down 58 Million by this extra trick and so just so spontaneously people started um uh going through the papers and and and finding uh know very very briefly holding the world record for the the BR of gaps uh the G the BR of gas primes and So eventually uh myself and and several other people organize this into an online um project which we call a polymath project as a play on Word polymath means someone who is good at many many different many things but here we mean it to mean many mathematicians poly mathematician working together uh each sort of sort of focusing on one aspect of of of J&#39;s argument and and some doing some numerics and other people adding some some theoretical advances and so we worked uh for uh online for uh several months uh and we managed to improve his bound I think by July 27th we managed to get the bound down to about 4,000 and then at which point we began to get stuck uh and we started writing up the results um so okay we&#39;re all very happy with that um but actually before we had finished writing up uh actually it turns out was almost 160 Pages worth of stuff actually calculations that we had to actually write up um it&#39;s just been trimmed down a bit but um but before we finished actually we found um we were surprised that um independently of our work um but probably inspired by by what y Tang did a postto at at Montreal his name was James mayard and the time here was uh November 19th found um a shortcut that simple that went uh that cut out a lot of the difficulty in jang&#39;s arent Jang used quite a bit of Firepower uh just give you one example uh there&#39;s a very famous result of delene um uh proving the wean hypothesis over finite f uh and it won him the fields medal and the Apper prize and so forth uh and Jang used that um to uh to get this um and we also did as well um but Maya found a shortcut that that that cut around a lot of this and uh so he managed to find a much shorter argument which of of the same okay but he got a much better bound of 600 actually okay um so after that came out you so we my group got very excited and we started talking to Maya and so he decided to joined forces with us and so we we worked a little bit more to combine our two methods and so uh we worked on that for quite a while and we eventually got uh yeah so that the the final thing that we uh that we declared victory at uh was back in April okay so our bound is now 246 and it&#39;s possible with lots of computer power to maybe shape another four or six out of this but uh but it it got really complicated and if we thought this is a good place to stop now um this this was a I mean um for the purposes of an online project it was really good to have this number as sort of a score right because you could you could it was a very tangible measure of progress which is something that you don&#39;t often get in a math in a mathematics project uh I should say though that this there&#39;s nothing that special about these numbers um you know I mean it&#39;s it&#39;s um what&#39;s more interesting ultimately is the techniques that go to proving these things and the various advances in technology that you need to get get from one to the next uh but still it&#39;s nice to to have this this this very clean progression here um if uh um but unfortunately you know I mean this looks like it&#39;s approaching too but uh slowly but um we we we unfortunately we do know that that these methods cannot um attain the tri time conjecture so um the most um we what we can prove now is that if you assume um so all these results are proven using the methods of C Theory which I&#39;ll talk about later um but um and there&#39;s a certain conjecture in C the which is kind of like the r hypothesis um something called the hem conjecture I&#39;ll explain what it is a little bit later well roughly um but it&#39;s it&#39;s sort of like a generalized ran hypothesis for C theorists uh and if it is if it is true uh we can we can improve this bound all the way to six that we can get infinitely many many pairs of primes six apart um so pairs of primes that are two apart are called twin primes uh pairs of of primes that are four apart are called cousin primes and pairs of primes that are six apart are called sexy primes uh so this would mean that there&#39;s either infam twin primes INF cousant primes or INF sexy primes so cute phrasing it uh in yeah um but unfortunately um we know also that this is the limit of the method that there is no no purely C theoretic method that can that can lower six to anything smaller um so if you want to actually get all the way to two you have to use something different from what anything that we do here um okay so this is this is uh uh these are some of the results we have we have other results as well um so one way to think about these results for example this result here is saying that you can find infinitely many intervals of length 600 which contain two primes um it turns out that that some of the later methods starting with mainard and and going forward can um also uh find um intervals which contain many primes um so so not just so which which means um so rather than than talk about prime gaps you can also talk about you know p n plus 2 minus PN and there there&#39;s some Bound for this and there&#39;s Bound for PN plus 3 minus PN you can find intervals of bounded length that contain three primes four primes and so forth um and this this this will become somewhat relevant later on uh but there there are other um variants of this and so the methods that been used to prove this uh these Prime gaps results have been also been used to prove many other things in number Theory um there&#39;s actually now know this is only like like 12 months ago but there&#39;s already like like a dozen papers using variants of what we now call the main ariv um or may not talve by some people okay to um to to prove other things okay so this is the state-ofthe-art for small Prime gaps uh now let May briefly talk about large Prime gaps um okay so large um a large Prime Gap is the same thing as a big long string of consecutive of composite numbers okay if you a whole bunch of composite numbers then the prime before and the prime the first Prime um to the left of that string and the first by that string has got a big gap so uh one of the first things you can prove and this is something some sometimes you even see in high school is that the prime Gap can be infinitely large so so this Gap can be arbitr large and there a very simple proof given any number n you take as factorial and you look at this consecutive string of numbers all the numbers from n factorial plus 2 up to n factorial plus n let&#39;s say n is at least two okay and these are all composite numbers because this guy is a multiple of two which which is bigger than two so it&#39;s composite this guy&#39;s M of three which is bigger than three so it&#39;s composite and so forth so youve got this this whole string of of um composite numbers of length n minus one so that means that that the primes to left and right of of this string must have a gap of at least n minus one maybe even n and n can be as big as you want so the prime gaps can be as big as you want now we can do better than that by using some slightly more advanced technology um so um okay so one of the first basic themes that we can prove about the primes maybe the most important thing about the primes really it&#39;s the prime number theorem proven in 18 uh 990 something I should know this okay but uh okay I think 1896 or 1897 um okay so uh the prime number theorem tells you roughly how many primes there are up to any given uh level so tells you that the number of primes up to a large number X is roughly X on log X and this is a natural log of x okay and roughly what you mean is that you actually have to multiply by a little error which goes to zero in in relative sense so the um that the ratio between the number of primes up to x and x log X goes to goes to zero goes to one as X goes Infinity okay so for a large number X like a trillion the number of primes up to a trillion is roughly a trillion over natural log of a trillion so that&#39;s how many primes there are in this interval and so just from the original principle okay uh what this tells you uh is that um among the primes so among the primes up to X there is at least one prime Gap um PN which is bigger than log X maybe a little error okay um or another way of saying that is that uh is an equivalent way is that the prime Gap can be as big as uh small factor Times log PN the prime Gap can be as big as the logarithm of of of the of the prime of say the first Prime infinitely often okay so this is a an easy consequence of the prime number theorem and the um and the pigal principle okay so um but okay but can you do any better okay can you make the prime can can the prime gaps be any bigger than this so um we don&#39;t actually know so much about how big the prime can be um I think the best unconditional bound we know is that the prime gaps are always at most a constant times PN to the I think 0.55 um this is so they can&#39;t be incredibly huge like like they can&#39;t be bigger than than than um uh 0.5 um basically almost the square root of PN this is a theorem I think of of hoo um no bam and pins this one um okay but um if you assume if you assume the r hypothesis uh you can get a slight Improvement uh we know that this that you can improve this to PN to to the one half and Times log PN the 055 yes yes it&#39;s it&#39;s a uh just a numerical constant coming from optimizing all kinds of other things um yeah but um if you the we hypothesis we can bound the prim up to the square root of of PN basically but that&#39;s still quite far from this pigeon hole um and this is not expected to be the truth um but this is just the best we can do even assuming this very heavy heavy machinery here which is the which is the r hypothesis um so the um the conjecture that everyone believes is J to Kramer so uh he used uh he introduced a random model of the primes uh so so he he modeled the primes so the primes up to X are just some set of numbers up to X okay and we know how many numbers there are but uh of course knowing the the number the cardinality of the set doesn&#39;t tell you everything about the set it could it doesn&#39;t tell his distribution uh so so krer had the idea well let&#39;s pretend that um this set we know is cardinality let&#39;s pretend that it looks random so he he modeled the primes by a random subset of numbers up to X of the same cardinality of cardinality approximately X of log X so in a way it&#39;s sort of giving up on the question of of what patterns there are on the primes let say I don&#39;t understand the set of primes let&#39;s just believe it&#39;s it is a random set of course the primes themselves are not actually random you you don&#39;t have to roll any die or flip any coins to to create the the P numbers but um the modern way of thinking of of Kramer&#39;s model is to is to say that the primes behave sto randomly that they behave like a random set even though they&#39;re not ex exactly random um now the reason you want to do this is because uh we have this uh probability theories are very good at Computing what happens to random sets and and we can compute all kinds of statistics of random sets uh so for example if you take a random set of numbers up to X of this cardinality you will find a lot of twins a lot of pairs of numbers two apart um quite a lot actually and so you uh this this gives you some evidence some htic evidence that t Prime conjecture is true um if you take this random subset uh and you look at Prime gaps between now these sort of fake primes these sort of random um model of the Prime what you find is that the the prime gaps and now the this this random variable they&#39;ll be distributed like a proon random variable uh whose um um uh sorry an exponential random variable um yeah the set looks like a Plus on process but the the gaps have this exponential variable of of mean log X uh and so you you you can get gaps larger than log X but they get exponentially rarer um you know as as you make the Gap bigger and bigger and um by Computing the random um by Computing the random um uh model krer conjectured that the correct size of the Gap should be log PN squ okay that if you make a sufficiently for a small constant C for espicially small C okay um that if you make this constant C small enough you should have gaps that are as large as the square of logarithm infinitely often but then he also conjectured that if you make the constant big um this should only happen finally often um okay he maybe he didn&#39;t quite conjecture that but this this is this is what the uh uh this is what the same model would would would predict for you okay that&#39;s that the largest gap should be roughly of the order of log squared okay you you get an extra log over what the the basic equal principle gives you okay but this this conjecture looks really far from reach um so I mean the the Trin Prime conjecture I I can believe that we just one or one or two clever ideas away from that you solving this um I mean we know C Theory by itself can&#39;t do it but maybe we just need one or two more clever ideas and we could maybe do this conjecture um this I don&#39;t think we&#39;re anywhere close to to um to solving um one of the reasons is that twin primes are still fairly numerous um at least we believe them to be fairly numerous among all the Primes so it said that the number of primes up to x is by X over log X um the random humanistics that come from this crma model suggest the number of twin primes up to X is about X log squ X U which is not which is still quite a lot of quite a large number there&#39;s a lot of twin primes and you just need to find one twin Prime in every say inter between X and 2x for large x um it it doesn&#39;t look uh completely impossible but with this uh with the large Prime gaps the larger you make your Gap the fewer and fewer um Prime gaps you have with um with that large gap well Le at least that&#39;s what we believe and and the density decrease exponentially so for example um we um yeah there should only be like one prime Gap this this huge in any interval between X and 2x or something like that um and so these are very hard to find uh these these huge large these huge gaps so I don&#39;t think we&#39;re anywhere near as close to this conjecture as we are to the CL conjecture although this one is also we are still quite far from that too unfortunately but we have some progress okay so it turns out between log log X and log and log squ X it turns out to be a surprising amount of of intermediate room um at least to a number theorist so those are sequence of results and it&#39;s it&#39;s a um puty well the list is somewhat amusing um so the first nontrivial result was by uh I can&#39;t pronounce the name polish mathematician West zinthus in 1931 who managed to beat the positional principle who&#39;s the first person to beat the positional principle and he showed that you can get Prime gaps that are as large as log PN times a correction term and his correction term turned out to be uh log log log PN except he could couldn&#39;t quite get that so he had to divide by log log log log PN okay uh there is a joke not a very good joke but yeah what do you what did the drowning number theor say and the the answer is log log log log log log log um but anyway okay so he was the first to get um uh to get to get some non true your improvement uh this was quickly improved well not so quickly but by by erish in 1935 I thought this is true infinitely often of course okay um and the same result uh but now slightly less embarrassing logs okay two logs on the top and uh three logs on the bottom squared okay um and then Ranken 938 uh managed to to to uh to improve this just a little bit more he could get um um Prime G as large as a constant times log PN okay again two logs over three log squar same bound okay and then he um was able to add an extra fourth log up here okay and C was I think one3 when in his first paper okay so you know okay so there there was there was some progress here um but amazingly what happened afterwards was that this uh uh people got stuck at at this at this Bound for for for uh for over 70 years um well so what had so over the next few decades people were able to improve c a little bit but they could not get um get past this this funny natural expression of of of of a combination of logs um so yes so C started steady improving there were results of uh um Sean hauger and and then ranking again May and pomerans and then finally pins until until this year um the most recent uh improve result on this was was this the same result just C was a bit bigger C was actually two * e to oiless constant which is a number about 3.56 something okay but other than that there was no further um there was no gain to this in fact this was so um stubborn and so mysterious that um erish who was fond of of of of giving out cash prizes for his problems actually offered $5,000 to anyone who could actually improve um to show that this constant could be taken arbitrarily large which is the same thing as as basically adding some some extra log or something to to this uh to this to this expression here um so this this is uh this is where things were stuck at for some some time um but then there&#39;s a a breakthrough this year and the funny thing about this subject sometimes is that um sometimes the time is just right for for uh for a breakthrough you know so mayard for example uh uh was was working independently of Jang on on on Prime gaps and the people were stuck for 10 years and then there were these two results which are pretty much independent of each other May was already working on this problem before Jang um and so we now have these two new methods of proving B gaps and the same thing happened here with large gaps so um uh so very recently there was a result of Kevin Ford Ben green green C Kagen and myself and uh this was uh 20th of August um okay so what we proved is that you can take C arbitrarily large okay so so this is true okay that you can get a tiny improvement over this um and amazingly uh the next day um this has happened to be before not not actually once it actually has happened 24 hours apart uh but sometimes I think just the the ideas in the air um so James mayard had released a proof from the next day proving the same result um by a different method and yeah so it was truly independent and actually the amusing thing is that he used results that we used to prove small Prime gaps to also to prove large Prime gaps which is which is an ni symmetry um so since then uh We&#39;ve joined forces and um we can now um um actually push this a little bit further so so um as of I think last week uh we can so all together uh well if you put everything together what we can do is that we can get rid of one of the log log logs on the bottom okay yeah so you can see we&#39;re not really getting all that close to clus conjecture of log squar um this this is these are still quite small improvements um and and there is a serious limitation to our methods we&#39;re not going to get anywhere close to to to CR this actually we believe is the limit the new limit of of our methods you will need a new idea to get to get Beyond this particular um thing um so yeah but but still it is nice progress and it uses um some of the theory we had before what would to oh um this year year yeah yes yes Ben actually announced the ICM in Korea uh okay okay so uh these are the results um so let&#39;s talk about some of how we prove things let&#39;s go back to small gaps okay so um okay so we have the the pigeon principle gives you something okay so the same pigeon argument that that gives you large Prime gaps also gives you gives you small Prime gaps so again the prim number the that for any large x the number of primes less than X or let&#39;s say between X and 2x okay if you take an interval between X and 2x the number of primes in this in this interval is is basically X on log X um and so again from the original principle um okay this tells you that there exists a prime Gap in the inter between X and 2x of of length Okay of the log X or less okay so just from Counting because the total the sum of all the gaps adds up to X and there&#39;s about X over log X of them so just from the pral principle there has to be one slightly one gap which is less than equal to the average spacing okay um so but but we want to do better than this we want we want a bounded um Prime Gap not not a gap that goes that goes logarithmically so um all right so how can you proceed well I can rewrite this argument so this is a very simple argument pigal argument but I can rephrase it in problemistic terms so uh problemistic reformulation okay so um so what I&#39;m going to do is that I&#39;m just going to find a prime um um I will I I will um look for a prime randomly so what I will do is that I will um Pick N pick a natural number and uniformly random between X and 2x okay so just pick some some number uniformly random from X and 2x it&#39;s a random number it may be prime it may not be prime it will probably not be prime but there is some chance that that it is prime you can confuse the number the probability that you&#39;re Prime and um from the prime number theorem um there are about X numbers between X and 2x if this if x over logx is prime the the probability that that n is prime is roughly about about one on log X okay so if you pick a number at random you&#39;ve got about One log log X chance of of it being Prime okay that&#39;s not very big but it&#39;s something okay um similarly um if you look at the probability n plus1 is prime if you take a random number X and 2x and you add one to it that&#39;s basically another random number between X and 2x not quite but essentially it is and so again the prime number the also tells you that the probability that this is prime is also about one in log X and so on and so forth okay so you can start you can you can take um you can shift your random number by various shifts 0 one two up to some shift say Capital H and H is going to be of size about log X okay okay so you can take your your random number and shift it um H times and each time you shift it you&#39;ve got another event that this shifted number might be Prime and there&#39;s some probability about one in log X that this is prime now if you pick H to be of size just a little bit bigger than log X you can make all these probabilties sum to be bigger than one so you add up all these probabilities individually they&#39;re small but if you keep adding them together they eventually get big so for for about the size you can make this the total sum of this guy okay you can make the total probability of all these events exceed one once you you make Capital H H big enough and about log X okay and the moment you do that the probabilistic positional principle tells you that if you have any set of events and the prob the total probility add up to to um to more than one then somewhere with positive probability two of these events happen together okay they can&#39;t be destroyed so somewhere for some n there must be two shifts n plus H1 n plus H2 where H1 H2 are of the size where they&#39;re both Prime and that gives you a prime gap of size less than log X okay so this is a very complicated sounding reformulation of the opal principal argument uh but this is a good reformulation and the reason that is good is because it generalizes okay so um this argument is very slick but it&#39;s sort of hard to see how to generalize it um this argument um okay so so first of all um rather than than taking um consecutive shifts you could you could be maybe slightly clever and take targeted shifts you know may shift H1 H2 you can you can shift your random number by by some well chosen shifts and as long as you can make the total okay and uh okay and this this probability is equal to something this prob something else as long as you can make the total probability of these guys being Prime add up to more than one you can and if these hes are not are not too far apart from each other you can create a fairly small Prime Gap from um from so you can try to play with with different hes um although if you pick n uniformly at random uh terms no matter how you shift you always get one of a logx and so you can&#39;t do much better than this log X bound that I already said but the um the but the more relevant Thing Once you once you phas things this way uh is that once you face things publicistic you begin to see that there&#39;s really no reason that you have to pick n uniformly um that uh that that maybe if you pick n not uniformly at random but some other well chosen probity distribution maybe um you could pick a better distribution from num between X and 2x such that you&#39;ve got a better chance of of making these um these guys Prime and so that you can increase um each individual probability and then maybe you need fewer of these terms is some to be bigger than one and maybe much less than log X okay so you know for example if you want n and n plus 2 and N plus 4 to be Prime and so forth um picking n uniformly at random and hoping that you get a prime is kind of silly you should at the bare minimum only search um the odd numbers if you want n or n plus 2 N plus 4 to be prime okay um that already doubles basically all the probabilities here okay so that that already gives you um gives you a win the price you pay though is that the hes have to become further apart you can&#39;t take consecutive numbers you have to take consecutive even numbers um but that already gives you some some uh some idea that you can play some games here that rather than work of a uniform distribution maybe work of a uniform odd number or maybe a uniform odd number which is also co- Prime to three or something like that and so you can start um so um so phrasing things in a in a non- probalistic manner um basically uh you have a freedom to choose a probity distribution now so so now the the new goal is to find a prob distribution or equivalently basically a series of Weights a sequence of positive numbers for n between X and 2x such that uh if you um so use these weights to uh to create a prob prob distribution by normalizing by the sum okay so um so you you you some these up you get some number and uh then you you sum up um all the ends for which say n plus H1 is prime okay so this ratio the the probalistic interpretation of this ratio is this is the probability that if you choose n using this weighted uh this particular set of Weights as a distribution that that that you get a prime when shift by H1 and then you shift by so if you shift you do all these shifts and if if you can addable these shifts and you can get this this thing bigger than one um this this implies that for some n at least two of the N plus h Prime okay which and if the H is the hi you the shifts are fairly close together you can get a fairly narrow Prime Gap in this manner um also you know if you can make this this ratio so big that is bigger than two then not only can you get two of these guys Prime you can get then get three of these guys Prime and so on and so forth so that sort of brings us to other things I mentioned before okay so um so basically all you need to do is that you need to find a clever choice of weight um which on the one hand you want the total sum of the total mass should not be too big you want this denominator to be small but you want this these numerators to be big uh you want uh um you so you want to find B what&#39;s called a Civ which which U which catches a lot of primes that um that that relative to Total size the portion of of of of the sum coming from primes is is relatively big much bigger than one of a log X okay so one just has to pick um this this weight properly um so um so the game then is is to find a very clever choice of of weight which does this so you know um so you could try the the uniform weight but that that that doesn&#39;t work very well um you could you could be overly clever and and um for example if you wanted to maximize a public that say n is Prime and N plus 2 is prime you could pick n not to be uniform uniform random you could pick n to be a uniform random among the twin primes in from X to 2x so from X to 2x you pick all the twin primes um and then of course you&#39;re guaranteed if you do that you guaranteed the N is that probably n is prime is one n plus two is prime is one 1 plus one is bigger than one and you win now why why doesn&#39;t that solve the problem um the problem is that uh that corresponds to to to to selecting new event to be one when N is a twin Prime and zero otherwise um and there the problem is that you can&#39;t make the you can&#39;t prove the denominator is non zero unless that&#39;s a hardest original problem um so if you want to make both of these probabilities exactly one uh we can&#39;t do that but we don&#39;t need to make to get anywhere near that far okay we um you could you could you know if for example you could find a weight where the probility N is prime is at least 6 and plus two is at least 6 that&#39;s already good enough um and so you can you can hope to find um um smarter choices of um of this weight which um uh which you can actually for which you can actually compute these sums without having to result to unproven conjecture and then hopefully you can make these numbers big enough that they sum to bigger than than one and then and then you you can declare Victory um so this is this is ultimately what everyone did so this strategy was introduced by goldon Prince in yum about 10 years ago um and that&#39;s how they got their first partial results on on on Prime gaps um now it turns out nowadays it turns out that um yeah so the question is is what weights do do you pick um and so on the one hand you want the weights to always be positive because otherwise um you don&#39;t get a proba distribution on the other hand you want actually the weights have to be simple enough that you can actually compute these sums um and it turned out after a lot of Tri and error that the the correct choice of Weights that the best choice of Weights that that we have currently come from What&#39;s called the silberg C CH by Atley silberg and the 7s I think um and so uh his weights um yeah so if if you want n plus H1 and N plus if you want to make a Civ such that all these guys have a high chance of being Prime um the soic Civ what what uh is uh what you do is that you take you take all you take your all these numbers that you want to be Prime and you multiply them all together and you get one big number and you look at all the divisors of this big number um and then you take some sum uh you take some sum of this um of this um um uh you over you sum some coefficients based on the sum and these are positive or negative coefficients and you take the sum and then you square it okay but you don&#39;t take all you don&#39;t take all factors you take all factors up up to a certain level so there there&#39;s some parameter r that you pick so you take all the factors of this big number up to some level R you you choose some weights and and so you get to pick what weights what weights you have here um and then you square this okay so the reason you square this is to guarantee it&#39;s positive so so these guys can be negative but the but the square is always non negative so this is always non- negative um and this turns out to be a fairly tractable expression to to to compute you know so um for example if you want to compute um U of n okay uh if you um if you unpack this this is equal to uh if you square this out this is equal to like the sum of all pairs D1 D2 up to R then there&#39;s some weights another weight and then you&#39;re summing Over numers N between X and 2x uh one subject to the constraint that D1 and D2 both divide n plus H1 n plus H2 so this is just expanding things out okay and the um the point is that this is something which is actually fairly easy to compute so if you just summing all the numbers from X to 2x this is basically just X um here you are summing all the numbers um n such that this product is divisible by by D1 and D2 so in other words it&#39;s divisible by the greatest the least common multiple of D1 and D2 now um if D1 D2 are bounded by R this number is bounded by R squ so as long as you pick R small enough like R less than say root X if you pick R small enough the modulus here is less than x and so this constraint is is a periodic constraint whose period is less than the interval in length of the interval here that you&#39;re summing over and that turns out to make this the sum very easy to compute uh and then and then what you get is you get some quadratic form in these coefficients okay and quadratic forms are something that that we don&#39;t scare us too much um if you instead of summing all of these guys you&#39;re summing um for example um this sum weighted by the the fact that this guy&#39;s Prime then you you get a very similar you get a very similar sum over here but rather than summing one you&#39;re summing primes uh the indicative function of the primes and so um what this sum is is basic basically roughly speaking this is counting how many primes there are in in arthic progression okay and there are lots and lots of theorems about what primes do in Artic progressions that that you can use some an analytic number Theory so that&#39;s so the the number theoretic input is then um to to control this use facts about about the number of primes in arithmetic progressions um so you may know for example dur theorem that if you have any residue class a mod Q where a is co-prime to q that this residue class always get INF minum primes um so that fact by itself is too weak to actually say much here but there are stronger versions of D theorem that that can go into here for example if you know gr you can plun it in here um but even without gr you can there are various ways you can you can compute this this expression and so you get another quadratic form so this this this funny ratio here what it is with the sober because a ratio of two quadratic forms with with with coefficients that you can pick and we know how to optimize quadratic forms right then you can do some linear algebra and you um and um you can start um you can start doing this and so um it turns out that if if you if you plug if you run this machine using um uh using the best theorems about primes andic progressions that were already available before yeang jang&#39;s work you you just barely fail you can get this ratio as close to one as you wish but you can&#39;t quite make it equal to one which which is very annoying uh so um go and P were not able using this method they so they introduced this method they were not able to get B gas but they they got really really close and the problem was that they didn&#39;t have a strong enough theorem about primes in arthic progressions um and jang&#39;s big breakthrough was that he found a new theorem about primes and E progressions that was not available before um uh oh okay maybe I I won&#39;t actually say exactly what it is um but um if you want to count the number of primes up to X in an arthic progression like a mod Q um this is this turns out to be a fairly easy task if Q is small but when Q gets bigger and bigger um the progression is sparer and sparer it&#39;s harder and harder to count the primes um and the standard tool in C theory is something called the bomber vinag theorem which allows you to to count primes in Artic progressions as long as this the space in Q is not too big basically as long as Q is less than x 12 um as long as you don&#39;t make the spacing any bigger than than square root of x we have a a very good theorem that tells you what the primes do in most of these arthic progressions um and and what Jang did was was was to be able to stretch this this suum a little bit and also say just a little bit about um primes in arthic progressions which were spacing just a little bit bigger than X2 and but it was a tiny amount I think the amount he goes like one over 568 or something um but that was just enough to tip the balance to make to make this expression just barely bigger than one if K was big enough I think K be something like three million um and this is what how he got his his his 70 million bound um initially uh and then subsequently we were able to push this enlarge this Con further and and optimize other parts of the argument okay so um and J worked quite hard to get that that time this this little epon actually he use a lot of Fury analysis uh a lot of um this work of Del on exponential sums and an incredible number of applications that Kashi Schwarz inequality um but um anyway he was able to to to get that um so mayard later on found that you didn&#39;t actually need so he spent a lot of effort to to get to get this Improvement um but actually even without that Improvement using just the existing primum in arthic progressions um Maya was able to to get the same sort of result um and so his observation was actually very simple uh which is that this is this was not the um the U the only C you could you could pick so the standard C until then had been to multiply all the numbers together here and take the devices of this product and create a Civ um but then mayard and actually I realized this independent at the same time it&#39;s another instance where uh uh have simultaneous Discovery but um but what you realize is that there&#39;s there&#39;s a more General C you can pick here so rather than look at the factors of a single of the single product you can just sum of us the separate factors which in retrospect is a very simple idea we should have realized this earlier in fact it had been realized earlier this this idea um but the reason why it was not studied further I think was unfortunately the first paper um to analyze the Civ actually had a serious error in it um and so it was somehow discredited and people didn&#39;t pay attention to it until 10 years later um yeah so rather than than look at the factors of this big product uh you split up the factors into into into K pieces um so now you have this multi-dimensional sort of Weights here um but then you can actually turns out you you can run the same game and um you you have another quadratic optimization problem uh and it now in m in many dimensions rather than one dimension um it turns out that it reduces to a certain calculus variations problem that you have some multi-dimensional integral of a certain test function quadratic form of a test function and in both the numerator and denominator and you want to to maximize the ratio um and turns out that um that if you choose the right test function you can actually make this ratio much bigger than than uh than in the one dimensional Cs and this is how he was able to to to get his results where in fact he could make this ratio as big as he wanted he he can get get many primes in B inter not just one prime okay and then afterwards the polym project improv us further uh we we uh we got some experts in in quadratic optimization on board and they they they they started optimizing these huge matrices and uh yeah this is how we we got our B down a little bit further and we made some other improvements too okay so that&#39;s all I want to say about small Prime gaps so let me talk now about large Prime gaps so um as I said before there is a simple construction of large Prime gaps so lot finding large Prime gaps is basically the same thing as finding large consecutive strings of of of composite numbers okay and the simple construction is just take n factorial and then you shift it by the numbers from 2 through to n and this already gives you um a large Prime Gap um if you f around a Sterlings formula uh this construction does give you a prime Gap and it tells you that the the side the Gap here is about log PN over log log PN infinitely often this is what comes out Sterling&#39;s formula uh so that&#39;s not so great this is actually worse than what you get from the prime number theorem prime number the the P already gives you a log PN and you actually want one log log behind if you use this simple construction okay but you can do you can be a bit smarter than this infactor is a bit wasteful um there&#39;s a smaller version of INF factorial called n this called the prim orial the the prim orial of so the factorial of n is all the product of all the numbers up to n um natural numbers the prime orial of n is the product of all the Primes of to n so you just you only multiply the prime numbers forget the composite numbers um and the same construction you take the primordial of N and you add two to it add three to it all they have to add in to it you still get all these numbers are composite by exactly the same argument and this this Prim is a bit smaller um so n factorial is bit like n n this is more like exponential of v um and so if you use the primordial which is the smarter thing to do uh you you can recover the ponal principal bound okay so you you&#39;re at least not losing anything uh but you haven&#39;t gained anything yet either sh sorry what the shs in oh yeah you you use all the numbers here so um yeah you need a consecutive string of composite numbers to get a large Prime Gap if even miss one then um um then you um yeah okay so um so why did this work okay so you can generalize this um so another way to think about this construction is that what we&#39;re doing is that we if you look at this interval this block of consecutive numbers from 2 to n um you can cover this interval by residue classes like if you take the mulus of two okay so that crosses out some of these numbers you take the mulus of three so those are basically Theos and if you take the mpol of all the Primes up to up to n every um um each of these resue classes between them they cover this this interval this interval is completely sied out it&#39;s completely covered by these rest two classes every number from two to n is divisible by One Prime at least between two and n and it&#39;s because of that that that&#39;s what makes this um this whole string consist entirely composite numbers because everybody here divisible by some prime you shift it by the primordial which is divisible by all the Primes up to n and so everybody here is divisible by one prime up to n now if you&#39;re facing this way what you realize is that uh um more generally if there&#39;s any interval you take any interval and if you can cover this interval maybe not by um um the the zero resue classes but if you can find some congruence class mod 2 um maybe not call it a I know it&#39;s called C some Cong class Mar 3 and so forth if you can find some congruence class mod um mod two mod 3 mod 5 which together cover every single interval here then um you can shift it um by n factorial maybe not by n factorial but by some other number okay um you can you can use the Chinese remainder theorem to to find some number Y which is I think has to be minus C2 mod 2 minus C3 mod 3 and so forth if you can find so by the Chinese remainder theorem because these all different primes you can find some number between one and the primordial which mod P is minus CP and then if you shift this interal by that Y what you&#39;ll find is that is that every number in here is divisible by one of the primes up to n and you would still get a string of consecutive numbers so so really what&#39;s going on in this construction is the fact yeah so um is the fact that that this interval between two to n can be covered by resue classes from from of of primes up to up to n and so if you can find a longer um um a longer interval which can also be covered by primes up to n then you can get then you can start beating this result okay so now the question becomes another C thetic question what is the most efficient way if you have a long interval of of numbers what&#39;s the most efficient way to to take AR class mod two and model three so you&#39;re allow to translate each each one okay but you want to to slide them around in such a way that they cover everything okay so this becomes more of a commentarial problem so it turns out that if you use all the Primes up to n um so the way we prove out our theorems is that so using the prims up to n we we can cover all the numbers from two to n but actually now we can cover all the numbers from from um we can cover a slightly longer interval um so I think the uh the record currently um all right well it&#39;s not so important exactly what it is but but uh but we you can cover an interval slightly larger than n using only only only um only primes up to n and so so how do you do this um so um the way you proceed here is that you you sort of um you modify the C osines you you um you use the the the zero congruence class for most primes say all primes up to say n / two okay if you if you C out all the mels of of of all the Primes up to n/ two that already cuts out a lot of numbers um any number that has any prime factor less than two will will be killed uh the only thing that survives or first of all one will survive um that&#39;s not very interesting but um but basically the only survivors so the um survivors are mostly just the primes between I think n/2 and this big number let&#39;s go y okay so um the primes are basically the only numbers that don&#39;t have small prime factors okay um so if you s out all these numbers uh all that&#39;s left are the primes and we still have all the Primes so but we still have some primes left over we we we still have some residue classes between n over2 and n to play with to to to sort of kill off any F survivors um as it turns out strictly speaking we don&#39;t quite do this um um we s out this for most primes P um and then we we we randomize it some of the pr it&#39;s actually better to to pick random R classes rather than than zero uh so there&#39;s some there&#39;s some optimization here which I I will L gloss over but roughly speaking um using using some of of of of these resue classes you can you can get you can reduce just primes in here and then the question then becomes uh can you cover the remaining surviving set of primes efficiently by congruence classes um coming from from from these primes here um and and so what the problem boils down to is uh can you for given Prime P between say n n/2 and n can you cover the primes okay can you find um a congruence class CP P which catches a lot of primes a lot of times in this interval from Z to one okay but H istically if every prime or most primes p in this in this range for each prime P you can find some congruence class which Each of which captures say hits about 10 Primes from one to Y then um if there are like interal login primes here then between them all these kind should be able to hit maybe 10 10 and Logan um primes out so you and so the more primes that each that that that you can hit with each congruence class um the the more efficient you can make the siing process so this becomes the basic question so um when Ben green Kevin Ford C Kagen and I got interested in this so um this looked very similar to something that we had worked on 10 years ago Ben green and I um so 10 years ago we had studied arthic progressions in the primes and when we we in particular we were able to find nice long arithmetic progressions of of any length you wanted where everybody is prime okay so uh or in other words we could find a congruence class mod R which hit a lot of primes um now for in our first paper this R was just a number it wasn&#39;t Prime we had we had a subsequent paper in which we could you could you could basically Force R to be basically a prime or maybe a small model of a prime um so we used um our Machinery from several years ago on Counting things like progressions and primes to be able to to um uh to answer this question but it was quite complicated because we this these results were themselves rather non-trivial um but what James mayai did he seems to have a knack for always finding the simplest way to solve a problem um is um um another way of thinking about this question you know so what you&#39;re doing is that you you&#39;re taking CP you know you&#39;re taking a congruence class which is like a big athic progression and so uh what Ben and I would would doing is that we were spending a lot of effort to make all these numbers prime um but actually for the purposes of the saving um you don&#39;t need to make all these numbers prime in order to catch a lot of prime you just need to have many of these numbers prime and you know James from his work on small Gast region primes had had already found a way to find to produce lots and lots of tupos like this where not everybody is prime but you know maybe say 10 of these guys are prime okay so he could find many patterns where where large numbers of these numbers are Prime and and uh but but also but not all of them um and this so basically if you just call this n and you call this H1 and H2 and so forth it&#39;s exactly the same problem um that um the the problem of catching um lots of primes inside a congruence class is a special case of of of finding a Civ such that n plus H1 n plus H2 n plus H3 these are all have a large highing Prime so rather unexpectedly the same basic method which underlies the the recent progress on small gas primes has now been used to also create large caspan tries um so yeah that that is a very nice symmetry yeah so we&#39;re still seeing what we can do further uh with with these methods although I think for the purpose of small gaps and large gaps we&#39;ve pretty much reached the limit of what we can do with these ideas but there are other problems in number Theory which hopefully we can do something about okay thank you very much

Transcript for:Prime Gaps and Recent Progress

Transcript for:
Prime Gaps and Recent Progress