An amazing thing about 276 - Numberphile

can we go old school Brady and talk about a particular number yeah 276 276 we'll do a deep dive into some unsolved problems in number theory if you're up for it pick a number what about like eight eight I'd like you to tell me its factors or rather technically it's proper factors but all that means is it's factors or devisers depending on where you are in the world not including itself one two and four and we're gonna add them up do you want to do that in your head yeah seven hey so this is a process called the aliquat process this quat bit other words like quote and quotient all of which are to do with part of something and actually it's the first time I realized that quote like you do a quote from someone you're telling us part of their speech like and that's where the word quote is come from but also quotient which is a technical maass word for dividing things like dividing into different parts Ali I think Latin alas also Alias something other when you rename something to be something else so this is like the other parts so the alqua process is taking the proper factor of this number and adding up and you've got a number which is lower than we started with but doing an alqua sequence means doing it again so you got seven go again tell me the proper factors and I hope you recognize this is going to be quite quick yeah I mean that's just got one because we're not doing seven itself so eight went to seven notice it went lower and seven went to one and actually if we carried on one has no proper factors if we're going to Define that is not including itself so you get zero but we might as well stop there I'm saying that if you get to one it's kind of got boring at that point then that's the end of our sequence your number that you chose eight got lower and actually that means we call eight a deficient number and I don't know if that's a word you've come across before it certainly turned up on number file but whether you remember it like every number is either deficient or something else and I do you know what else could happen if you pick his proper factors and add it up I mean it could could add up to more than the number do you know what they call it like Surplus uh yeah that's a good uh synonym for the word I'm thinking of which is the word abundant abundant that's right I do so numbers could be deficient like or abundant we can try and pick an abundant number do you want to try and guess one um the clue is actually a number with lots of factors well let's try 24 because it is actually Four factorial but also like it's famously got lots of factors one two uh does three go in there it does yeah three factors are also used to useful to do in pairs so if you've got two you could probably tell me 12 because two * 12 and 3 * 8 is 24 4 * 6 there's the six I was thinking of and I'm pretty sure we've run out and actually factor pairs are quite useful you sort of have to head towards the middle 24 would go with the one but since we're doing proper factors we're not going to include it so let's just add these up 20 3 6 we're found an abundant number Brady well done Eight's division 24 is abundant and actually let's do the aliquat sequence because this time it's gone higher so if I take whatever answer I get and do it again let's go 36 36 has factors 1 2 and 18 three goes in and I think that would be 12 four goes in and that would be a nine with it six is a factor as well pretty sure those are all the factors can add these up I mean if this needs fast forwarding ready you know what to do 30 39 45 49 52 54 55 also abundant abundant but I wonder if you can predict what's going to happen now because this sequence I just keep doing it this is what an alqua sequence is and you just see what happens so 55 uh 1 * 55 2 three no none of those go in pretty sure it's 5 * 11 and I think that's it doesn't feel like it's going to be abundant anymore 11+ 5 16 17 which is Prime and I think you know what happens to primes you found one earlier it goes to one game over here so this one took one two three terms or fourth terms to get to one I'm not really too worried about but it looks like even when you started with an abundant number this thing eventually hits deficient and starts tracking back down to one if it's not abundant or deficient what else could it be these are rhetorical questions because these are called the perfect numbers ah lovely yeah and so six is one of them if you take the proper factor six one two and three you add it to get six and if you see if you do an alqua sequence on a perfect number it's never going to get to one unlike the other ones we've looked that they will stick in a loop 28 is famously perfect the next one is 496 and they turn up and there's all sorts of stories we can tell why we never found an odd perfect number I'm not getting into that as interesting as it is what I like about these aloot sequences is they are doing things that I already knew quite a lot about perfect numbers abundant deficient but now there's another question which is if you keep doing this process what could happen is there a number that never goes to one well we've seen some that do go to one so we know it can do that we've also talked about some numbers which don't get to one because they're perfect what else could happen and I think there's some other familiar landmarks we just haven't spotted yet on the horizon we'd have to be talking about numbers that when you sum their proper factors go into a loop couldn't they just keep being abundant abundant abund there's the other question um could you find an aloot sequence that doesn't converge so let's first of all establish the the loop thing famously 220 if you add up its proper factors you get 296 and if you add up the factors of 196 the proper factors you get to 120 in fact let me just get my keys uh so I tot you want to find my my bunch of keys I don't at me right they're too big I know but this is the number 220 uh this is called an amicable number because it is best friends with 296 those two go in a loop and if you do the Alo quart sequence of either them you just get the other one and this is half of a heart uh it's horribly cliched romantic to say that my wife has the other half of my heart and she has one with the number 296 on this is the lowest pair of amicable numbers that exist okay so there there's the there's the loop what about forever so the question is uh and this is now a famous piece of mathematics called the Catalan Dixon conjecture do all aliquat sequences either end in one or hit a perfect number and loop or amicable numbers and in fact there are other cycles of what they call sociable numbers which don't go in an amicable pair but they go in Loops uh and there's a whole other branch of number theory about that but the Catalan Dixon conject is do they all go into a loop or hit one and this is where you stop doing it by hand because it's easy to check the low numbers and it becomes tedious to check the large numbers so we're going to bust out some code okay so what I did was program joj to take the proper factors have them up and then spit it back in and all it's going to do is plot a little graph of where it goes so what we're expecting to see is abundant numbers start with the graph going up and anytime you see a tick up in the graph we've found an abundant number and division will go down perfect stay the same and you're kind of stuck there so let's just do a quick check here is the number six I think you know what's going to happen I've told it to give up after like six repeats so if I click go on this the number six doesn't change it's a perfect number your number that you started with was eight so let's go on eight and we're expecting it to drop straight down went to seven went to one and it's gone to zero and stopped working let's try the other one you did which is 24 and we're expecting this time to to go up a bit better then it comes down so now this I love about this sort of number Theory thing this is basic maths but we've got an infinite number of numbers to explore and some of them are surprising so if we do do one that we just talked about 220 there's that amicable pair bouncing around let's try some other ones I'm going to try the number 95 because I know what happens and this has got a special name this is called an aspiring number uh and I hadn't heard that term we've got a lot of these familiar sort of friendly number words like perfect abundant deficient amicable sociable aspiring is one I hadn't come across but here's one of them so here's a number that doesn't get to one actually got to six and it's stuck there so 95 is not perfect but it's almost there so it's aspiring to be perfect that's I think why they've named it it knows the way to perfect it knows the way to perfect that's a good definition of an aspiring number if you're using an alqua sequence and again I was like why have I I heard this word and I actually asked a bunch of respected math math communicators and mathematicians and some of them have heard it if they've done the number the they're like of course I've heard of it and a bunch of us including myself were like I've not heard this and yet I know all the other numbers that come out of this little study so I enjoy like doing something new so 95 is aspiring that was a new one for me let's try some other numbers I'm going to try 30 so 30 look at that I mean I'm sounding excited because actually you try loads of numbers and they're deficient most numbers are deficient and they just shoot down and end at one 30 goes up for quite a long time it jumps up to the number 260 and then it's all over and it collapses down but here you have a first Glimpse that some numbers go quite high before they come back down 25 what you reckon 25 ends up being aspiring cuz it goes to six straight away uh and then it stays there so let's try a more surprising number uh you don't have to go very high to hit some surprises I'm going to try the number 138 if you are remembering the number I mentioned at the beginning just hold that thought because it is connected with this uh and I hope that my joj programming skills let me get away with this up and up and up and up oh down oh and up oh it's playing with you oh it's a yes go son go never stop oh no no no no oh no it's back it's back I mean I've deliberately slowed this down to give you the chance to commentate so well it's gone so high now we can't see any of the blits earlier amazing oh no it's down again it's down my computer is chugging away doing the calculations here oh no it doesn't look good but we saw it did it was looking like that before come on come how long should we watch I mean if it's still going it hasn't found a perfect number it's just that we can't see these numbers they're so small compared to the 18 billion that was up here and at this point it hits one it stopped oh 138 hits one but before it gets there it hit almost 18 billion hold on let let me check right so thousand million billion 180 billion uh you do not want to do that calculation by hand what a ride that was and I I enjoyed your commentary cuz the first time I programmed this and the fact that it went slowly enough like if it was all over quickly you wouldn't have seen any of those flips and you still can't see them now right Ben you know what we should start a YouTube channel where we just pick a random number where we just pick numbers and then commentate the sequence let's do it like marble racing we'll do some b-roll footage of that in a minute but um the other problem is this is a nice observation about like you can't see the journey that fun time we had is vanished in the in the week down here right because this number is so big that these are minis School in comparison so this is a prime candidate for not plotting this on a linear scale we're going to use a logarithmic scale and I'm just going to flip over to the log scale on the screen and what that will do is instead of plotting the number it will plot the basically the power of 10 and that number how many digits that number has so I'm going to use log base 10 to plot it and I'm going to flip straight over to that there is the journey we went on on a logarithmic scale himas yeah and we've got this mountain range and all that commentation going on here like this is 180 billion so logarithmically there's a huge difference to down here but a log scale helps you see these things it's a really nice example of why log scales help you see detail when there's more than a few orders of magnitude going on but 138 did collapse eventually so it now feels like a genuine question are there any that don't come back and the answer is we don't know of course not but there are a few interesting examples so first of all let's hit the headline and then we can go exploring the jungle again if you like um if you double 138 you get 276 which I may have mentioned earlier um I'm going to run that one on python for reasons if I run it on joji uh the thing melts right joji is not a good processing thing for doing heavy duty calculations and you can imagine like in this one it had to factorize a number in the hundreds of billions now it can do it but it's not quick and I hope all number of our viewers are aware that factoring numbers takes a long time when they get big and that's partly why the internet is still secure hooray uh python is better at doing this stuff so I'm going to run this on python I don't know if I can take the excitement of another one like that okay some badly written python script here I'm actually going to run 138 again so we can see how that looks here it comes right that's it doing the number crunching that like we did on paper that's the aloot sequence it reached an end which means it's going to start animating it and now you can see that mountain range we were commenting on is already doing a log scale by the way because I knew it was going to be impossible to see and the reason this is working more efficiently is that it's done all calculations first and it's just done a little animation for me so I'm going to do this for 276 and you'll see why immediately I needed to do on python let's just see what happens so here come the numbers they're spitting out the bottom of my screen they're getting big right it's given up it's got too large for today down here and it gave up after 70 and already like we're up at 70 steps 70 steps and it's gone beyond 10 the^ 13 but it could crunch to one in like a step it could and I've gone a bit further and so have others and the reason I'm telling you about this one is that we have not gone far enough to know and we've got the supercomputers involved this has been running on people's computers for a long time it's the first number that we genuinely don't know everything else up to 276 we've checked and it's either collapsed one or got in a loop uh and we'll see some nice examples and we've done loads of other numbers too much higher and a lot of them all claps to one and go in Loops but this one surprisingly low if you ask me we just don't know and we do not have the computing power to get to the end of it and it hey The Next Step that someone calculates you're right it might collapse we might hit a deficient streak but these numbers you could hit a perfect number well you could we don't maybe this is an aspiring number like maybe this one wants to be perfect and we'll get there eventually but we don't know the Catalan Dixon conjecture says maybe all of them go to one or hit a perfect Loop but this is a counter example if it doesn't we just don't know how to check if it does CU we haven't got the computing power so here's an unsolved problem with matics and what I like is this is low hanging fruit it's within that territory that a lot of us know about with with recreational number Theory perfect abundant deficient numbers the 276 is the first of what they call the Lamer five he was a mathematician he found five numbers less than a thousand that we don't know about are they all still open or they all still open so there's other numbers less than a thousand that also we don't know about but they tap into the same sequence so for example if you try 306 it does some stuff and then it slots into the same numbers that 276 does so you end up with exactly the same graph in the same shape so they feel like less unique but there are a few of them but there are five with unique patterns that are not coming down as far as we've checked under five and they are 276 552 564 660 and 966 and we just don't know do you want to see what some other numbers do that we do know about yeah right let's let's explore the jungle a bit because there are literally infinitely many numbers to explore and the ones we have come across some of them are nice all right first up for your consideration I'm going to do 98 0 460 go go gadget aliquat sequence the numbers will spit out first well one divides 220 2 four five uh 10 what I like about this is that they missed a pair in fact they missed the second smallest pair of amicable numbers and these were found 200 years later it's never going to hit five what we don't know well at the moment five is the only Untouchable number we found which is odd all of the others are even so unsolved conjecture number two of this video is five the only odd Untouchable number

Channel: Numberphile

Views: 227,561

Keywords: numberphile

Id: OtYKDzXwDEE

Length: 15min 38sec (938 seconds)

Published: Wed May 01 2024