Untouchable Numbers - Numberphile

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 980 460 go go gadget aliquat sequence the numbers will spit out first and off it plots right and the reason it's stopped is it's detected that it's done a cycle of things the same so it's hard to see on the scale but it's actually found an amicable pair I think it's found the pair 220 and 296 so I don't know actually whether there's a name for this it's aspiring to be an amical pair I don't think it's called an aspiring it's romantic cuz it because it likes the idea of finding it has romantic aspirations yeah maybe so but you can noce how high this is and like the log scale these are numbers in the sort of 10 to the^ 9 sort of region and it has to chunter around up here before it comes back back to numbers which we kind of recognize so it's just nice that you can see they have to go super high and then you end up with different outcomes let's do another one 2856 still not as low as that weird one we don't know about but 2856 feels like you know under 10,000 so let's see what this one does and again these are log scales oh why did you say oh it's like it's hit a mega loop it's hit a mega loop that's a good word for it although Megan might employ that it's got like more than a million it's only got 28 in that Loop okay some people call this the ECG graph it looks like a sort of cardiogram heart rate thing has found a cycle of sociable numbers of of length 28 uh which I didn't know that even existed but this alqua sequence is tapped into it and so it has to go quite High into the 10 to the six sort of area and then it starts going around the mega loop as you call it here's a question yeah what's the biggest sociable Loop they know so now we're into a lot of open questions but I think Loops of any size are out there or Loops of unbounded size there are some Loops which don't exist I don't think there's a three cycle hey but these are things I'm quoting without checking if you want to go and check a good web page for this the al.de is a fairly sort of low budget but really brilliant resource for people who are exploring this and they're keeping the latest research coming out for which ones have they settled the lam of five are still not settled but there's a bunch of other stuff that people are running on computers at home and you could do the same if you write your own basic python script better than mine would be a good idea and you could check I'm sure there are efficient ways of doing the really large numbers that I haven't bothered to implement I'm imagining this moment that someone's checking one of those Lamer fives and suddenly it drops and hits one and it's like jod Foster hearing the hearing that thing in contact yeah detecting the rhythm of the universe like oh my goodness I need to call someone and tell them it has that that feel about it it's like digging into these irrational numbers but this is just number Theory this is like properties of numbers which we didn't put there and this is why like some people are bound to ask rightly so what's the point why I explore this stuff and I think there is no reason other than it's there to explore and if you if you don't get a tiny glimpse of the romance of just I want to know because it's there and I didn't put it there and none of us put it there but it has these properties and I think it's worth remembering that that's what number theory always is and it's only ever really done because we're curious as human beings it does turn out number Theory to be pretty useful some time to time let's do Internet Security cliche again prime numbers factorizing them is hard and that's how we keep things secure but that's not why we do this stuff if you're this far into a number file video about alot numbers and you're asking what's the point you're definitely watching the wrong Channel there's a lot more to research and I I guarantee that you'll have a fun time if you've enjoyed this introduction go and read some of the websites that are out there and see what research is being done right now about this stuff all right as a PSA resistance Brady I thought I'd write a python script to do all of them and when I say all of them I don't mean all of the numbers but let's say like up to 500 cuz actually most of the numbers you check drop to one quickly and the ones that are difficult like 138 like take a bit of computing power 276 is going to fail but as long as I put a cap on like stop looking now we should be able to do a whole sve of them and I could plot all of them for you if you want to see that bring it on we we'll plot them all um it's going to take a little bit of time and you'll see all the numbers track through so here we go these are all the alot sequences anytime you see a long one um you you got an interesting number it's up to 230 right 276 is just on the screen we're expecting a pause it gave up it's now carrying on my code is pretty efficient if I do say so myself it's it gets stuck a few times though and you see some lawn sequences in here but hopefully we'll see a graph of all of these and I'm going to plot them on a log scale we'll see some of the lay of five turn up and they'll be like unbounded ones here we go and all these things are sequences end in a in a finite number of steps so there's like some sequences join up and these ones like you got these parallel lines I think that's the 138 and ones like it that tap into the same sequence and we're expecting them to end eventually but they go quite high and there's a few other things while it's animating through you might want to speculate about what are you seeing here I think I've only let it go up to about 150 terms I was lying clearly longer than 150 oh yeah nice parallel one's crashed yeah so that's the 138 and and its family that tap into same sequence all of them go super high but actually they're doing the same pattern there's two parallel branches going up looking like quite similar that's 276 and like the other one I said that goes with it there's a couple of the other lame of 5 in here I think 552 and 560 and you can see some other things the perfect number down here another perfect number here we've got some amicable pairs in fact both of them the 220 and 296 are in there that nice zigzaggy pattern and I think that's 496 another perfect here but I really like this graph as a sort of summary of all the stuff we've been chatting about because you can see some of the expected Behavior with familiar words like perfect and these ones that we just don't know about and it feels like we're not even close to knowing about we just don't know how to check as far as we need to check maybe they come back down and maybe they don't I'm beginning to think maybe they don't so the Catalan Dixon conjecture is that everything goes to one all gets in a loop with a perfect number or sociable group but the the counter conjecture guy and Selfridge now have put out there which is they believe that there are unbounded sequences and they reckon they found heuristic evidence that it's not a proof and they are the first to admit that but they found lots of evidence that even if you keep checking it seems less and less likely that we will be able to prove they all come back to one that's their claim at the moment I don't think it's been solved and it's current number Theory research as far as I'm aware I'm out of my depth at this stage but that's a nice feeling for an Amon mathematician to sort of some find this low hanging fruit that gets you into deep water quite quickly and if for viers want to go and play with alqua sequences I recommend it I've enjoyed it well what I enjoyed about doing alqua sequences I learned some new terms which I feel like I should have known one of them is an Untouchable number which had not come across but is only defined in this context so an Untouchable number is apparently a number that will never ever be in an alqua sequence ever for example five I believe has been proven it will never turn up in any Al sequence so all those lines we saw bouncing around is 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 retire convinced that if you have a m n Prime 2 to the nus1 and you then multiply that by 2 to the nus1 you always always always get a perfect number and then they will go home and I will say well how do we know it works how do we know that's always going to work we have we haven't proven that

Channel: Numberphile2

Views: 125,692

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Id: Yh1QUYn2f3I

Length: 8min 9sec (489 seconds)

Published: Wed May 01 2024