Okay today I will tell you about this very fascinating number: 420514
upon 7, that's the fraction. Now these are a special class of sequences called Somos 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 1s. You can predict that the algorithm is simply a_n+1 is a_n, a_n
equals 1. 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 a_0 is 1 and since it's Somos-2 which means that a_1 is also 1. If you
want to compute a_2 your sequence right now looks like this: you have 1 and 1. a_2 will be a_1 times
a_1 divided by a_0 which will just be 1. Now look at what happens for a_3: you get a_2 times a_2
divided by a_1, which again is 1. So, you can believe me that you only get a string of 1s.
Now the sequence is not very interesting but the algorithm is getting juicier. Now let's go to
Somos-3. The algorithm is a_n is a_n-1 a_n-2 divided by a_n-3. Our seed is going to
be a_0 is 1, a_1 is 1, a_2 is 1. All right? So your sequence is starting from three 1s, now a_3 is
a_2, a_1, a_0. This is 1. - (Brady: I'm getting pretty sick of 1s!) - Well you need to wait slightly longer.
So you have a_4; now, since you're already sick of 1s, you'll see another 1 because you have a_3
a_2 a_1 - so sorry, you have 1 again, and 1 again. It'll continue to be a string of 1s.
Now you see these are pretty boring at the moment, so the statement is that the first three terms
of this this different Somos-k sequences, which means that Somos-1, 2 and 3
are all just strings of 1s. Nothing interesting happening but it just gives you a feel of where
the algorithm is going. Now we go to Somos-4, will we see something interesting there is the question?
- (I hope the answer's yes) - So the algorithm is a_n is - now I'll write it as a fraction - numerator
is going to be a_n-1, a_n-3 plus a_n-2 squared upon a_n-4 with a_0 is
1, so it now has a string of four 1s in the beginning. - (And what numbers does that spit out?)
Okay so a_4 is going to be a_3, 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 a_5? If you do a_5 it's going to be
a_4 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 a_5 a_3
a_4; now you could say that I'm cheating because all the denominators have so far just been 1,
you'll say that a bit longer because a_2 is also 1. What is this going to give us? 3 plus 4 divided
by 1 and you get a 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 2 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 314, 1529 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 - I have earned your trust. You have Somos-5. So you just start with
five 1s. Now you can see what's happening, if it's Somos-4 it starts from four 1s,
if it's Somos-5 you start from five 1s. And this is very important because if you don't start
with all 1s 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 1s you need to get you always need to stick to that one. It'll be 2 and 3,
but it won't be the same as Somos-4 of course; you'll get a 5 and it'll go to 11, 37, 83, 274,
1217, and it goes on. - (Integers.) - Yep always integers. So Somos-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 Somos-6 - again, all integers, no fractions.
Somos-7, all integers, no fractions. Somos-8, something happens or not? That's the thing, okay
I'll just write it down. Somos-8, it starts with a string of eight 1s, and what is
the algorithm? You have a_n-1, a_n-7, a_n-2, a_n-6, a_n-3, a_n-5
a_n-4 squared divided by a_n-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 1s up to this point, so
what's the next digit? Ah, it is 4. Now you go on, you have a 7, you have a 13, 25, 61, 187, 775, 5827,
14815 - and this is a_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 16th term times the
10th term, 15th times 11th, 14th times 12th, 13 squared divided by 9. Which is
this term, the last one, times 13. The next one is 5827 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 420514 divided by 7, and it's not divisible by 7.
- (So it's a fraction!) - Exactly! The rule has been broken! - (Ah no, we got so far.)
(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. - (So the other ones went- all the ones before went all)
(the way to infinity without a fraction?) - Without a fraction; but in the Somos-8 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 - 420514. We'll give this a special box,
because it's the first fraction it deserves its own box, and then you got 28670773
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. Somos-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 changed)
(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.
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. 420514 over 7.)
The troublemaker? - (Yeah troublemaker number!) - Or the trendsetter?
(Oh yeah, that's true as well,)
(yeah, it's a more positive spin.) - The trend-setting number for all these sequences.
(Nah, I like trouble-maker.) - 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.