The Troublemaker Number - Numberphile

Key Takeaways

okay today i will tell you about this very fascinating number four two zero five one four upon seven that's the fraction now these are a special class of sequences called somo sequences and they go as follows this is really a very fascinating story in general it's called somos k sequences where k can be a number so let's say you start from somos 1 it's just a string of wants you can predict that the algorithm is simply a n plus one is a n then equals one it's very very simple nothing magical is happening here it's very not interesting now somos 2 i'm going to give you the algorithm and we'll look at the sequences and there is a pattern in this in this algorithms that will that will get clear in a minute so once i start writing enough of these you'll go ah that's the algorithm and you're starting from this and you're saying a0 is 1 and since it's almost 2 which means that a1 is also 1. if you want to compute a2 your sequence right now looks like this you have 1 and 1. a2 will be a1 times a1 divided by a0 which will just be 1. now look at what happens for a3 you get a2 times a2 divided by a1 which again is 1. so you can believe me that you only get stream of months now the sequence is not very interesting but the algorithm is getting juicier now let's go to almost three the algorithm is a n is a n minus 1 n minus 2 divided by n minus 3. rc it is going to be a 0 is 1 a1 is 1 a2 is 1. all right so your sequence is starting from three once now a3 is a2 one a zero this is one i'm getting pretty sick at once well you need to wait slightly longer okay so you have a4 now since you're already sick of once you'll see another one because you have a3 a2 a1 so sorry you have one again and one again it'll continue to be a string of ones now you see these are pretty boring at the moment so the statement is that the first three uh terms of this this different somos k sequences which means that's almost one two and three are all the strings of ones nothing interesting happening but it just gives you a feel of where the algorithm is going now we go to summers4 well we see something interesting there is the question i hope the answer is yes so the algorithm is a n is now i'll write it as a fraction okay numerator is going to be a n minus 1 and minus 3 plus n minus 2 squared upon n minus four with a zero is one so it now has a string of four ones in the beginning and what numbers does that spit out okay so a4 is going to be a3 a 1 plus a 2 squared a 0 1 plus 1 divided by 1 which is 2. oh we got a new number exactly now what's happening with a5 if you do a5 it's going to be a4 a 2 plus a 3 squared upon a 1 3. so things are starting to move still integers though we're still getting still integers now look at this there is no reason that this should give you integers right so let's do another one see if we get a fraction yep that's that's the idea you have a5 a3 a4 now you could say that i'm cheating because all the denominators have so far just been one you'll say that a bit longer because a2 is also 1. what is this going to give us 3 plus 4 divided by 1 and you get a 7 7. we got 4 strings of 1 that was already given and then you have 2 3 and 7. now let's go on and maybe i will introduce you to the first term which has a denominator that is not 1 so that you know i'm not cheating and maybe now you can take my word for it that the next one will be 23. a 8 will be 69 plus 49 divided by 2. so you have 118 divided by 2 which is 259. you see even when the denominator is two you still got an integer no fractions yet no fractions yet now this goes on this goes on for the whole sequence that you'll never get a fraction right and the next one is three one four one thousand five two nine and so on it's doing two things it's not giving you a fraction but the number is also growing it's growing pretty fast okay now you can say well that's a bit too much of a coincidence okay one can check it and there is good reason that it's not giving you a fraction okay now i'll introduce you to another non-trivial thing and i won't go through the computations but i'm guessing you can believe me at this point yeah i have earned your trust you have so much five so you just start with five ones now you can see what's happening if it's almost four it starts from four once if it's almost five you start from five once and this is very important because if you don't start with all ones then you do get a fraction but then it's not really somos it's not really the magic of all these sequences so if you want the magic stick to the rule that first however number of ones you need to get you always need to stick to that one it'll be two and a three but it won't be the same as almost four of course you'll get a five and it'll go to 11 37 83 274 1217 and it goes on integers yep always integers so it's almost 4 all integers 5 all integers so what's really happening it's curious enough that it's always giving me integers but then the question is how far is it going to keep giving me integers does it break is there some god-given rule what's happening so so mo 6 again all integers no fractions so 7 all integers no fractions it's almost eight something happens or not that's the thing okay i'll just write it down okay so most eight it starts with a string of eight ones and what is the algorithm you have a n minus 1 a n minus 7 a n minus 2 a n minus 6 a n minus 3 a n minus 5 a n minus 4 squared divided by a n minus 8. so you see the algorithms are also growing they're growing very nicely but there's a nice pattern going on there these are all ones up to this point so what's the next digit ah it is four now you go on you have a seven you have a 13 25 61 187 775 5827 one four eight one five and this is age 16. still all integers still all integers clearly i know what i'm doing so what is the 17 digit okay let's do it so you have the sixteenth term times the tenth term fifteenth times eleventh fourteen times twelve thirteen squared divided by 9 which is this term the last one times 13. the next one is 5 5 8 2 7 times 25 so it's the term before the last one times 25. now you see the pattern you have the multiplication of this multiplication of this and the next one is 775 times 61 which is these two and then there is 187 squared which is sitting right in the middle all divided by 7 which is a 9 which is the guy here and that will equal 4 2 0 5 1 4 divided by seven and it's not divisible by seven so it's a fraction exactly the rule has been broken ah no we got so far yeah that was the fraction you showed us at the start of the video yeah and that's the first fraction that appears in these sequences ah so the other ones when all the ones before went all the way to infinity without a fraction without a fraction but in these almost eight sequence you get a fraction is that the only fraction no that's the interesting part now you can continue and you get so let me write this down four two zero five one four we'll give this a special box because it's the first fraction it deserves its own box and then you got 2 8 6 7 0 7 7 3 divided by 91. fraction not an integer and this goes on you get quite a lot of fractions it's not very clear if you'll find an integer somewhere but up to the first 30 digits or so you never get an integer again all right okay so this is where it breaks yeah so what happens with somos 9 and 10 and 11 same thing you'll think you'll get integers in the beginning but you'll eventually see fractions and there is an interesting thing also about which term is a fraction it's almost 8 the 17th term breaks now what happens for somos 9 it is the 19th term and for 10 it's the 20th term so it's almost twice as whatever the number you're getting here what happened what change that made it break all of a sudden when it was doing so well and always staying as integers that's a hard question there is quite a lot of deep understanding of these the algorithms that i wrote down those are very very important for all sorts of things so for example the somos 5 algorithm which was this guy this comes out as some discrete integral system that's the buzzword uh so there is a good reason why these are just integers but i don't know why it breaks at a higher order but when it doesn't break we have some understanding of what's behind them so there are these things called cluster algebras which are associated to them there is all sorts of very very nice mathematics that is also very recent so that's one of the reasons that these are super exciting that's exciting that number there 420 514 over seven the troublemaker yeah troublemaker number or the trendsetter oh yeah that's true as well yeah yeah it's a more positive spin yeah yeah okay the trend setting number for all these sequences now i like trouble okay so there is this constant somewhere out there in the world of numbers with this formula which will give you every prime number in order as well not missing one