Today I want to talk about the online
encyclopedia of integer sequences. This
is a website which has- it's a
database full of sequences. In fact the
the encyclopedia has been going
for nearly 50 years. It started in
book form and it's now grown into a big
website. So it started off by a guy
called Neil Sloane and he- for his
research he needed to- just needed to
tell some sequences apart and he wrote
them down. And he just started to collect
his sequences together, they were on to
punch cards - this is going back quite a
while now. And when the internet came
along that was a nice opportunity to put
it online and there are now two hundred
thousand and more examples of integer
sequences. So there might be things like
the prime numbers, the square numbers;
just sequences with special properties.
The classic example, right: 1729 is a
important number in mathematics, it's
part of mathematics folklore. 1729 was
the number that Ramanujan talked about
when Hardy asked him, you know, this is a
boring number can you think of anything
interesting about this number? And he
said 1729 is the smallest number that
can be written as the sum of two cubes
in two different ways. It's equal to 1 cubed
plus 12 cubed and it's equal to 9 cubed
plus 10 cubed, so that's an interesting,
that's interesting isn't it? Now, in fact there are
other interesting things about 1729,
that's not the only interesting thing
about it. And so it actually turns up in
other sequences - I'm just going to pull
out- this is the paper that I'm going to
refer to, in here it gives 1729 as an
example. So let's have a look,
1729 is also the third Carmichael number,
that means 1729 divides x to the power
1729 minus X
for all X. So whatever value X is it will
divide that perfectly - it's the third
Carmichael number. It's also the sixth
pseudo-prime number in base 2; which
means 1729 divides 2 to the power 1729
minus 1, minus 1. So there you go, so that's
called a pseudo-prime. It's the
thirteenth number that is one more than
a cube. And here the last example they've
got here I quite like this one: 1729 is
the total number of ways to express 33
as the sum of six integers. Again all
these different properties for
1729. The point
is, the more properties it has the more
sequences it appears in and I guess the
more interesting the number is. So the
more often it appears in the
encyclopedia of integers the more
interesting the numbers should be. So what
the guys who wrote this paper have done
is they counted how often each number
appeared in the encyclopedia of integers.
And they looked at what pattern that
made. Now, try and think of what pattern
you think that might make, but this is
what they got. And you can see it's quite
clearly a curve, it decreases, it's a
curve. They found a little formula just
to describe that curve: this curve - I'll
write it on the paper -
is 253 million divided by N to the power
1.33. So that's the
equation of that curve. So these numbers
are appearing far more often in the
encyclopedia of integers, these turn up
really often and then the ones that are
lower down here appear less often. But
here's the interesting thing; you can
quite clearly see in this curve a gap
appearing. There is a quite clear space here.
And yet the numbers are actually forming
two bands. So there's a top band which is
filled with
the popular numbers, they're turning up
the most often, then this is huge gap and
then underneath there's a bottom band
there which are your less popular
numbers, they're the ordinary numbers
that aren't turning up so often. And
that's quite surprising. They could
generate this sort of curve if they had
a machine generating sequences and
integers, this is exactly the sort of
curve you would expect, so that wasn't a
surprise. The surprising thing was they
have this gap separating popular numbers
from the less popular numbers. Now the
ones in the top band are actually
numbers like the primes. All the primes
are there, in fact almost all the primes
are in the top band. About 95% of square
numbers are in the top band, in the
popular band. There are numbers like
numbers that are just perhaps 1 or-
plus 1 or minus 1 off a power of 2;
so that'd be 2 to the power n plus 1 or
2 to the power n minus 1. Or there might
be number such that have lots of factors;
numbers that factorise into lots of
little factors, those tend to appear in
the top band. Other numbers that appear
in the top band are ones that have
special properties in base 10; so numbers
like 1111 are in the top band, 2222,
3333 all appear in the popular band of
numbers. So what they think is happening
here is obviously this is cultural - that
the popular numbers are the ones that
the mathematicians like. These are the
primes, or numbers that are related to
primes, and they are they are elevated
above the ordinary numbers and so they
have a special place all by themselves.
And they're elevated by such an amount
that you can see a quite clear space
between them.
We'd like to thank audible.com for their
support of this video; and if you go to
audible.com/numberphile you can
actually download a free audiobook. Now
there are loads of mathematics based
books we could recommend, but today I
thought I'd recommend something a bit
different: The Fault in Our Stars by John Green
which is an excellent book. I think it's
probably targeted at a slightly younger
audience than me
but it's a book I really loved. And
John's also a really great member of the
YouTube community and makes awesome
videos. The book's also just been made
into a film so you might want to get
ahead of the game before the film comes
out. So again, thanks to Audible for their
support of our films and if you want to
check them out and maybe download that
free book: audible.com/numberphile.
So in 2012 the smallest number that
didn't appear was
13,794 - doesn't appear
in the encyclopedia at all. Now I
went and checked if that was still true
and I'm afraid that it is no longer
still true, I think it's about four
sequences that that now appears in.