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Strong Law of Small Numbers - Numberphile - YouTube
Channel: Numberphile
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Are some numbers more interesting than
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than other numbers? And I think they are.
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(Brady: So like an interestingness rating?)
- Yeah, I
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think there are in in the following
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sense; there's actually a paper about
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this. So the numbers that are the most
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interesting are the small numbers. Now
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this might be counterintuitive, because
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obviously there are large numbers that
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are interesting like googol, Graham's
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number, Googolplex - obviously we'll talk
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about the videos that I've made. So there
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are big interesting numbers; but there's
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actually a theorem, a sort of a theorem
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it's a joke paper, by a guy called
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Richard Guy, who's a pretty well known
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mathematician. And he's tried to
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formulate the the Strong Law of Small
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Numbers. And what he says is that
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small numbers are too interesting; as in
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they they're sort of, you know, there's
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just not enough of them to cope with the
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demands that are put on them essentially.
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Now why is this? Why does he say this? He
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says, well basically you look at small
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numbers and you look at sequences of
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small numbers, and you can be tricked. You
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can be tricked into seeing patterns
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which then don't extrapolate to very
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large numbers; or the converse of that,
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you can be tricked into seeing things
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that don't happen which actually you
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know maybe sort of you know some some
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characteristic behaviour. You can
be tricked
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into seeing things that don't happen
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which actually, when you go to very large
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numbers, it might be some sort of generic
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behaviour. But looking at the small
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numbers is just not enough. And he's
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formulated this in a- in this
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paper here 'The Strong Law of Small
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Numbers'. It's a great little paper, it's
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just quite a fun paper to look at. What
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he does is he comes up with a bunch of
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examples that sort of back up his his
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claim. It's done tongue-in-cheek, in fact
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he says this is proof by intimidation
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which is I think is a brilliant line.
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But anyway, let's just- I thought
it was worth
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looking at some of his examples
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to see how small numbers can trick you
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into drawing erroneous conclusions. So
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the first example is he looks at numbers
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of the form 2 to the 2 to the n plus 1.
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Ok? So if I put n equals to 0 in this,
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okay, then that's 2 to the 0 is 1,
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2 to the 1 is 2,
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plus 1 is 3, so I would get 3
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out okay? If I put n equals 1 into it,
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well you can check you get 5 out. You
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put n equals 2 into it
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you get 17 out. If you put n equals 3
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into it you get 257. For n
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equals 4 you get- well this number okay?
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So, you know, we're going through,
these are a
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bunch of small numbers, smallish, starting
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to get quite big now; and
what d'you notice
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about all these? Well they're all prime,
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okay? So no less than Fermat thought that
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perhaps all numbers of this form would
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be prime numbers. But of course,
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along came Euler, probably the greatest
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mathematician of all time, and actually
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went the next one up, n equals 5 - so we
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go to n equals 5. So 2 to the 2 to the 5
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plus 1 turns out is equal to 641 times
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6 thou- well, some big number right?
So Euler was
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able to show that this was true. So this
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pattern that seems to emerge at the small
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numbers is just- you know, when you get
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big enough it just disappears. So it's
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sort of the small numbers have tricked
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you. There's another- so another
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example that I just picked out
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pretty randomly, another one I
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liked, was this sequence, okay? Again
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they're all related to prime numbers,
number
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theorists love prime numbers. Number 31,
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clearly prime, 331 is also prime. 3331
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is prime, 33331 is prime - you can see
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where I'm going with this right? Carry on
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like this, these these are all prime
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numbers, so does that mean
if I take a bunch
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of 3s and then a 1 it's a prime number?
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Well, you keep on going and it looks
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like they are, until you hit 8 of the
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3s. If you take eight 3s: 3333
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33331; so this is the first one not
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to be prime and this has been shown
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to be equal to 17 times 19607843.
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So, again, small numbers tricking you into
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into being something that they're not. So
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small numbers are almost too interesting
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and yeah, all numbers are interesting
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especially the small ones.
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[Preview] Five bilion, five hundred thou-
million duh, duh, duh-
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prime is, it actually gets a better
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result the bigger the number is.
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