63 and -7/4 are special - Numberphile

so what we're going to try to talk about today is the arithmetic of dynamical sequences let's do an example so here's an example maybe you'll recognize let's let f of X be the function 2x plus 1 and what I mean by iterating something under this function is let's start at a point say x equals 0 when we plug x equals 0 into this function we get 1 and then let's keep doing that so when we plug 1 into the function we get 2 plus 1 which is 3 when we plug 3 into the function we get 6 plus 1 which is 7 so you look at the last number so 15 31 so maybe I'll do two more just for a demonstration here so 62 plus 1 that's 63 and 126 plus 1 it's 127 so this sequence is special so the sequence you might recognize that each of these numbers in the sequence are off from a power of 2 by exactly 1 so this is known as the Mersenne sequence well people have heard of the Mersenne sequence because this is a famous question that we don't know how to answer is are there infinitely many prime numbers in this sequence since this question is so hard one easier question we could ask would be what are the prime divisors of each element of this sequence look like that seems like a harder question well it might be a harder question depending on how specific you want to get but so what I mean by the prime divisors of elements of the sequence is we have these factors so each number factors into prime numbers as long as we're ignoring 0 here so 3 is already prime 7 is already prime but 15 this is 3 times 5 31 that's already prime 63 well let's see so seven is a prime divisor and 9 is a divisor but it's not prime it's actually 3 times 3 times 7 and 127 let's see I think that's prime and so on anyway so this is what I mean by prime divisors and so instead of asking okay do we have infinitely many elements in this sequence which are prime we could at least ask as we go along in the sequence do we get a new prime at each step so that's the question I'm interested and answering so here okay we don't have any prime devisers but here three is a prime divisor that never appeared before here seven is a prime divisor that never appeared before here five never appeared 31 never appeared but here we run into a problem we broke the streak no new primes is that bad well depends on what you're interested in but if the question is do we always have a new prime divisor for the Mersenne sequence the answer is no because look at the sixth element doesn't have a new prime divisor but the interesting fact is after that all Merson numbers have a new prime divisor every single one every single one what happened with 63 well it's just when you're at the beginning of a sequence the numbers are so close and there's so many Prime's happening that it can occur that you sort of fail to have this new prime divisor and it also matters that this was the sixth element of the sequence but that's a little bit complicated so the point is that after that after that six element of the sequence after 63 we always have the new prime divisors so this is an interesting thing and I think the next natural question once you notice that a sequence like this has some interesting property is what's special about this sequence what other sequences have this property so instead of choosing a function like 2x plus 1 to generate my sequence what if I chose a nonlinear function so if I had an x squared in there so let's say I chose x squared plus 1 but we can still start generating these sequences so if we start with 0 if we plug it into this function is going to map to 1 again but when we plug one in we get 1 plus 1 which is 2 when we plug 2 in we have 2 squared plus 1 which is 5 we plug 5 in we have 25 plus 1 which is 26 and when we plug 26 in I happen to know we end up with 677 so because I'm dealing with an x squared instead of an X here the sequence gets larger much faster but still we get the sequence of integers still we can ask when do we start getting new prime divisors so if we look at what we have here so notice again 2 & 5 are already prime 26 is 2 times 13 and 677 is also prime and it turns out that for this sequence it's even better than Merson sequence all numbers after to have a new prime divisor so this has proven that after 2 you always get a new prime divisor in this particular sequence you showed me an amazing sequence then you showed me a more amazing sequence ok well then get ready so let's do the same thing except that there's no reason that we're only talking about positive integers here right so let's look instead at something like this and see what happens so again we start with 0 0 maps to negative 1 negative 1 maps to 1 squared minus 1 which is 0 but we already know where 0 maps 0 maps to negative 1 and so the sequence we get is completely not interesting that's cool that's more interesting well it's definitely not going to have this property that we have new prime divisors at least so this is bad news right as far as this question is concerned this is a weird sequence so let's let's try a different negative number let's try I don't want to try minus 2 and maybe you can try it and find out why let's try instead minus 3 all right so let's go so 0 & 0 maps to 0 minus 3 minus 3 maps to 9 minus 3 which is 6 6 maps to 36 minus 3 3333 maps to something messy that I don't want to calculate but now we'll get an infinite sequence again instead of having this this repetition like we had in the minus 1 case we'll get an infinite sequence and again this sequence will have new prime divisors at each point 0 seconds every time I'm beginning to think this is just special thing anymore I know so let me tell you you're sort of right so here I'll write down a general fact if say C is an integer so a whole number or the negative of a whole number and C is not 0 minus 1 or minus 2 mostly because we want to avoid this problem then every element of the sequence that's generated from x squared plus C has new prime divisor so the only exceptions are 0 minus 1 minus 2 this works for X Squared's but it works for X Squared's plus an integer if you wanted you could ask this for a fraction see instead of a whole number see so if C was 1/2 let's say so if f of X is x squared plus 1/2 well then what starts happening so 0 maps to 1/2 now the computation is a little less pleasant but 1/2 is 1/4 plus 1/2 which is 3/4 3/4 when we put it into this function we have 9 over 16 plus 1/2 which I guess is 17 over 16 does that sound right and so on so even though we're getting these fractions in the sequence the denominators are just powers of 2 and so when you try and factor them the only prime that you get there is 2 so we can still ask the same question about the numerators in the sequence so we're just going to we're just going to ignore them just ignore the denominators there's not interesting stuff there happening there anyways so if you ignore the denominators then we look and we have this sequence 0 1 3 17 and so on and it turns out that again you will have this property it seems sort of fundamental and part of the reason why is because I've been hiding an example from you so so maybe one last example of computation that I'll show you is this kind of tricky little thing here x squared minus 7 over 4 so what is 0 map to 0 maps to minus 7 over 4 21 over 16 minus 7 over 16 squared all of a sudden everything comes to a stop because look at our numerators 0 seven seven times three so three is a new prime divisor there but here we don't have a new prime divisor so there's something different between 1/2 and minus 7/4 for some reason there's an element in this sequence where the numerator doesn't have a new prime divisor and this is this the only this is the only number the only fraction that throws the spinner in the way we don't know we don't know so just to give you a hint of why this is true so this is the only one that we know of but we don't know if it's the only time it can possibly happen so what's really going on behind the scenes here has something to do with a particular set of numbers known as the Mandelbrot set which I won't go into but it's this interesting fractal where things go kind of crazy and it's hard to draw a fractal so I'll just do my best good thank you so let me put this into perspective for you so this is a picture of the complex numbers and I'm interested in the numbers that are inside of this set and let's look at the values of C that we've done this computation for so one we first did x squared plus one one is over here we also did minus one minus one is over here we did minus three which is well outside of this set we did one half which is also not inside this set but minus seven over four it is inside of this set so minus seven over four it turns out is right inside this piece of the Mandelbrot set this lon of the real of the real numbers there that's it seems to run right through us way the fractions that would be is that aren't there going to be loads of fractions that are going to cause this problem then that's a really good point so you're absolutely right there's there's tons and tons of fractions on the real line inside of the Mandelbrot set but it's not enough it turns out to just be inside the Mandelbrot set you have to be particularly very far inside of the Mandelbrot set and so this is really saying something special about where the number C can lie inside of the Mandelbrot set so if there are other numbers law - seven over four that are going to throw a spanner in our works they definitely have to be in here that's right so right away you get for free that if you take a large enough fraction say bigger than one for example although you could get closer for free you get that there will always be new primes and those sequences but yeah if we were to look for other examples of where this might occur they would have to lie inside of the Mandelbrot set and they would have to lie well inside the Mandelbrot set actually it turns out but it looks like there are numbers here that would be pretty well inside the Mandelbrot set or is it a particular it's a particular thing inside each of these bulbs here that I've drawn sort of half half heartedly there's a special point and what might happen the only reason why this might happen is if your rational number is very very close to that special point in a technical way that is is hard to formulate I think it hard people don't believe that there's one beyond minus 7 over force if I had to make a guess I would say probably not but it would certainly be nice to know that for sure

Channel: Numberphile

Views: 1,619,856

Rating: 4.9067922 out of 5

Keywords: numberphile, Mandelbrot Set (Namesake), prime numbers, prime divisors, factors, dynamical sequences, iterations, Prime Number (Field Of Study), mersenne sequence

Id: 09JslnY7W_k

Length: 12min 13sec (733 seconds)

Published: Mon Mar 24 2014