Yeah, so I'm going to write down a number
for you Brady: 14972.
Now, what's interesting about that
number?
(Brady: It's the subject of a Numberphile video!)
Well it is, but the thing about that number is it's
seemingly not interesting at all, okay? So
as of a a couple of days ago I've been
reliably informed that this is the
smallest number not to appear in any
sequence in the On-line Encyclopedia of
Integer Sequences. So this- this does
actually appear in a definition of a
sequence as a coefficient, in the
definition of the sequence, but it
doesn't appear in a sequence itself. The
first number not to appear at all is the
following number; so this is seemingly
even more uninteresting. So, okay, can
they be uninteresting? Well let's think
about this. And there's actually a
mathematical proof - in inverted commas - that
says that actually there is no number
that is uninteresting, that all numbers
are interesting. It's very simple; let's
prove it by contradiction. So let's
imagine that n- we're assuming that not
all numbers are interesting, okay? So if
not all numbers are interesting then let
the number n be the least uninteresting number.
Okay, so n is the least uninteresting
number. But now n is the least
uninteresting number so it's interesting!
Okay? So that implies that n is
interesting.
Okay, so you've immediately run into a
contradiction. Okay? So all numbers must
be interesting. Now of course this is a-
this is slightly dodgy, this proof;
this is- it's what we call self
referencing. It's a bit like statements
like 'can the Almighty create a
stone that that he can't move?' If he's
almighty he can do anything so he
should be able to create a stone that
he can't move but, you know, actually
then he can't move the stone. So there's-
(So he's not almighty?) - So he's not
almighty. So there's something sort of
self referencing in it, so what you
really need to do is you need to define
what you mean by interesting. So you can
imagine doing that; so you say, okay,
what's a- what makes an interesting
number? Well I'll write down a list of
criteria; so maybe a prime number is
interesting. Square numbers? Pretty
interesting. And you write down a list of
things that you think are interesting
and at some point on that list, let's say
at the number 28, say, whatever, you write
down that it's the least uninteresting
number as one of the criteria. But that's
clearly now- this is ill-defined now. So
the problem that you have here is is
that the criteria aren't well defined;
you need to draw the the conclusion
about a number before the definition is
complete, so it's it's fundamentally
flawed as a logical problem. But let's
come back to these numbers, okay?
(These two at the top?) - Yeah let's work on the assumption-
of course this is Numberphile, right? All
numbers are interesting right? So so
these must be interesting, right? So so
when I found out about these particular
numbers not appearing in the- in the OEIS
I thought, right, okay, there's gotta be
something interesting about them; so indeed
I found something. So I was playing
around with them and what I noticed was
was the following:
so let's calculate the cross sum.
Cross sum's when you basically take all the
numbers that appear and just add them
up, very simple. So in this case it's 1
plus 4 plus 9 plus 7 plus 2, ok? What d'you
get when you do that? Well you get 23. Ok, fine.
Now look at this one, I'll do it as well; 1
plus 7 plus 0 plus 8 plus 7. And you do
that sum and what do you get? I get 23 as
well, that's pretty cool. So, alright, they
both give 23 so that's that's cool. And
what did I notice about 23? Well, 23 is of
course a prime number, so then I thought,
well there's got to be, I bet you, there's
a sequence in that list that has
those numbers whose cross sum are prime - and
it's there. It's in the list, its
sequence, for anybody wants to check,
A046704. This is the
sequence of cross sums who are prime.
And so these numbers should really
appear in it, but of course they don't go
high enough for this number but really
they should be in it. I guess if they
were to do this properly these these
numbers would soon drop out so that's
proof I think that all all numbers
are in fact interesting. Anyway, so
just thinking
about this, you've got me thinking a bit more
about is there such a thing as a boring number?
Okay? And I found a definition of a
boring number online and there's-
somebody somewhere has defined a boring
number as a number that you can't
express in less than six English words.
So, for example, 999 - forget
the and - 999 is four words okay?
So that's not boring. So I was thinking
about it, I think the the first one that you
come across is 1121, okay. So this I
think you need six. But then I thought, oh
hang on a minute, what if somebody comes
along and says, oh that's actually eleven
hundred and twenty one, then I can get
it under. But I think 2121, I think
that one is pretty safe. Nobody is
calling this twenty one hundred and
twenty one, if they do that then then
they annoy me. So this this is definitely-
this I think is this smallest boring number.
But what's happened there? I've just called
it the smallest boring number, that's
that's four words. So I sort of fall
foul of my criteria again. Of course this
is all just just messing about.
You know, there's just not enough of them
to cope with the demands that are put on them.
It says, well basically, you look at small
numbers and you look at sequences of small
numbers and you can be tricked. You can
be tricked into seeing patterns which
then don't extrapolate to very large
numbers-