Squaring Primes - Numberphile

Key Takeaways

today we're gonna look at the fact that all prime numbers when you square them are one more than a multiple of 24 which a lot of people don't believe it you are shocked when I first told you about this you are you're beside yourself because people think the prime numbers haven't got a pattern like a lot of emails from people saying oh I found a pattern the prime numbers are like yes there are loads of patterns in the prime numbers and including this one which will test so Brady what prime number should we do with the black sharpie let's do 17 7.5 okay 17 squared equals okay so it's gonna be no I can do this cuz squares gonna be a hundred and seventy plus seven times that which is gonna be seventy plus 49 okay so just off the top of my head at work was that I'm sorry seven eight twelve thirteen okay I think it's roughly that's not right seventeen times 17 equals to 8 now I was right doubted myself and so I'm saying that's one more than a multiple of 24 and so we can split that apart because we've got 240 hidden in there plus 49 left over and that's one more than 48 which is a multiple of 24 and so this whole thing here is gonna be as 12 times 24 plus one so I argue any prime number you give me if you square it you'll get some mobs pull of 24 plus 1 on the end 5 5 is 25 that's one more than 24 you could have opened with that Brady but now we do 17 first it doesn't work if you go back up and do 2 or 3 so 2 or 3 don't work too small too small and I mean I argue that they're not real prime numbers I call them the subprimes so I would like to ignore those for the purpose of this they don't work everything 5 onwards this always works but we should prove it right you didn't take my word for these things so it comes down to when I first saw this first of all I was amazed and then I was like what hang on there must be a reason why there's this pattern is that there are loads of patterns in the prime numbers so there's a nice one involving multiples of 6 and because 24 is a multiple of 6 I was like you know what that might be something to do with it so I'm gonna recreate list out a number line 11 12 13 18 19 okay so we got those and then let's find our favorites the primes we've got the subprimes hanging out down here and then we've got the real primes there's five there seven there 11 13 17 19 and so on and you go right where are the multiples of six there's multiple of six there it's six look at that there's a prime on either side of it right so those two either side of multiple six there's the next multiple of six look at that there's a prime on both sides of it nothing else in between next mode will six over there look at that there's a prime on each side of it so the primes are always above and below all the multiples of six except doesn't always work so if I kept going the number layer stopped at 19 so 23 is a prime 25 is not a prime there is our next multiple of six and only on one side do we get a prime 25 is not up until now they have all been and the reason this one isn't is 5 finally caught up with us so 5 is prime but then every 5th number isn't tens not 15 s not 20 s not boom so that's not this one out the moral of story is not that there's something magical about the primes and they happen to always be above and below every mod for 6 it's just that's the only place they can be so here's a multiple of 6 and here's the next one they can't be at the even points in between because we know Prime's can't be even so immediately it knocks out these two and it can't be the number in between because multiple 3 number between each multiple of 6 is multiple of 3 so it can't be on any of these which is why it has to be above or below a multiple of 6 it's just a fancy way of saying Prime's don't have 2 or 3 as a factor and then to their constrain to those and so people are happy if you say all primes are odd ignoring the small ones right and that's just because this does nobody's saying Prime's haven't got 2 as a factor if you say all Prime's are 1 more or less than multiple of 6 and was like wow but all we're saying is they haven't got 3 as a factor and then when we say all Prime's squared give you one more than multiple 24 it's a variation on this it just looks more impressive and so the way I'll show you the way I first worked it out when I came across that I cuz like right I'm going to prove this delivery so here's what I did I said any prime number other than two or three is it either gonna be some molds pull of six plus one or some multiple of six minus one so every K we can put in one of these two categories and so my thought was I'll just square these and show they're both one more one less than a multiple of twenty-four but then they didn't work out really really complicated so I had another cheating moment when I realized this K here is either going to be odd or even so that K is either gonna equal let's use skews em this time to M or it's gonna equal this is or to M plus one it's either even or odd and so I can split each of these into their two options so this one is either gonna be if I put two M in there it's gonna be 12 M plus one or if I put in two n plus one it's going to be twelve M plus seven and then down here that's either going to be put into M that's gonna be 12 and minus one or it's gonna be 12 m plus six negative 1 5 okay so now I know every single prime number falls into one of these four categories and so then I went through and I took each of these and squared them to see what happens when you square now this is not exhaustive like an exhaustive proof where I've checked every single option I've just taken all the options and put them into four categories and now I'm going to check each of the categories separately so let's do them quickly M squared plus two times that time this one's 144 m someone will correct me if I'm wrong - tutors at times 2a plus 25 okay that's what we have to do now is show that every single one of these is a multiple of twenty-four plus one and the first ones reasonabie straightforward because that's a multiple of 24 because that's 12 squares that's 6 times 24 so we actually put 24 outside of that's gonna be 6m squared plus M plus one so multiple Oh 24 plus one this one is 24 outside of again 6m squared plus seven M plus two plus one so I've done theirs at 48 I've sorry that 49 I've realized is 48 plus one and so the 224 is the 48 plus the one at the outside yeah same deal again 24 outside 6m squared minus one plus one that's what mass m and finally 24 outside of 6m squared plus 5m plus 1 plus 1 so there I've taken every single prime showing that must be in one of two categories above or below modulo six each of those has to be one of two categories if that multiple six is even or odd those four categories cover every single prime I've then expanded them out and shown that if you square them you get some number times 24 Mobil 24 plus one which proves it right it's a slightly clever exhaustive proof where I put it in categories and I've dealt with the categories one at a time I was so pleased when I got to the end is nigh was like yep I knew it had something to do with being one or more one less than a multiple of six I did some algebra I worked it out I showed some friends of mine and one of them said why didn't you do it the easy way and I was like ah you know me I like to do some algebra easy way so it turns out there's an easier way to do this so I did this way this is mine I love it my friend Paul said look all you're doing is you're looking at P squared minus 1 and asking is that a multiple of 24 every prime squared subtract one is a multple 24 okay P squared minus one you may remember this from school or you could still be in school and you see that and immediate you think well that's difference of two squares its P minus 1 P plus 1 if it's been a while since you did this at school you can just double-check that if you multiply these out you'll get back to P squared minus 1 so what we really want to know is is this a multiple 24 well what can we say about this well on the number line that's gonna go over here we're gonna have the number 1 less than P P minus 1 here we're gonna have the prime P and then above it we can have the prime P plus 1 so actually we got three consecutive numbers here and the prime in the middle hasn't got any factors so we know every second number is a multiple of two so because these are in a row we know either those two are a multiple of two or that one's a multiple of two and this can't be a multiple of two we know P is not even because it can't have two as a factor so both of these numbers have to have a they're both multiple of two these are both even numbers in fact because they're two consecutive even numbers one of them we don't know which one of them is a multiple of four so either this one's a multiple of two this one's a multiple of four or that one's for and that one's too right do we know when we multiply them together the combined total will be a multiple of eight so now we know this it is a multiple of eight because it's two even numbers either side of a prime and one's for ones too we have to get eight now we've also got three numbers in a row and every batch of three numbers one of them has to be a multiple of three every third numbers a multiple of three again it's not the middle one because we know that when there is not a multiple of three because it kind of have 3 as a factor it's a prime so one of these has to be a multiple of three again we don't know which when we multiply them together the total must be a multiple of three so we also know that is definitely a multiple of three and if something is definitely a multiple of three it's definitely a multiple of eight it is a multiple of 24 and so that's it just because the two numbers are either side of a prime if you multiply them together you get a number which is a multiple of 24 and actually we've not really used the fact that this is a prime all we've used is the fact that it's not even and it's not a multiple of three so what we've actually managed to prove is that all numbers which don't have two or three as a factor if you square them you get a number which is one more than a multiple of 24 and that all the ones either side of every single six so what vaccinations approve is both all primes are either side of multiple of six and if you square any number on either side of a multiple of six you always get a number which is one more than a multiple of twenty-four and that is one of my favorite prime patterns you said that your friends one there was easier it certainly is prettier and it uses less ink but I'm not sure that would have been easier to have come up with that's very true so I'm using easier probably in a strictly mathematical sense where I guess easier I'm kind of using it to mean less turning of the handle arguably you're right this one is easier because there's not a lot of creativity I can say that it's my proof I can say it all I've done is just chunked it into predictable categories and then turn the algebraic handle and tidied it up and that's my response where's this one once you've got it is easier to follow but it wasn't easier to come up with so I guess I'm saying it's easier from looking at it in hindsight not easier coming up with it creatively in the first place both mathematicians love a creative proof right and so the more creative you have to be coming up with the proof normally the more impressive mathematicians consider that proof I like to think you're on numberphile we go pretty deep into our topics but I'm also aware sometimes you want to go even deeper really dive in today's episode sponsor the great courses plus is superb for that these on-demand videos cover everything from yes mathematics through to other things like looking after your dog or playing chess your teachers are going to be experts from all over the world leading universities places like that and there are over 10,000 video courses to choose from now when I'm going through the site I've got bit of a weakness for videos about Egypt Egyptology this one here decoding the secrets of Egyptian hieroglyphs this is definitely one to have a look at it's presented by Professor Bob Brier who's one of my favorite egypt explainers he's great and he's gonna have you writing in hieroglyphs before you know what's happening although somehow I doubt that's the actual rosetta stone behind him I certainly hope it's not now for a free trial go to the great courses plus.com slash numberphile that should be written on the screen beneath me and there's also a link down in the video description where you can find more information oh by the way that prime number squaring stuff you just saw a map talking about that's just one page among hundreds in Matt's book things to make and do in the fourth dimension I'll also include a link to that underneath in the description