Sometimes it's useful to have a number which has lots of factors.
Like the number 12 which can be divided by 2 and 3 and 4 and 6 and 12 itself.
We find these things useful sometimes.
The ancient Greek philosopher Plato
He thought the best choice for something like this was the number 5040.
He thought this was the best choice because it had loads of divisors, lots of factors.
So, 1 divides into it obviously.
2 divides,
3, 4, 5, 6, 7 divides, 8 divides, 9 divides, 10 divides
So all the numbers up to 10 divide,
12 divides into it as well,
And it's got 60 divisors all together.
So Plato was thinking this is the best number you could have for, like say, a city.
If the population of a city was 5040 you could divide that up into all kind of different groups.
If you wanted to divide up the land
Then you'd divide it up into units of 5040.
5040, lots of divisors, plus it has more divisors
then all the numbers less than 5040.
We now call this a Highly Composite Number.
[Brady Haran] It's like an Antiprime
[James Grime] It's like, yeah, it's like an Antiprime
it's the most divisible number, lots of divisors
In fact the first guy who really properly studied this
was the famous indian mathematician Ramanujan
did it about 100 years ago
Let's take a look at properties of Highly Composite Numbers.
Antiprime Numbers
(laughs) I don't think it's gonna catch on Brady, I think you are 100 years too late for that!
So the definition of a Highly Composite Number
is one that has more divisors,
factors if you wanna call it factors,
more divisors than any number smaller than it
let's just run through the numbers, let's find some
so these are like the previous title holders
yeah, exactly!
How many divisors for 1, it's just one.
So, 2, it's a prime, so like primes do, the only things that divide primes are 1 and
itself it has two divisors, primes have two divisors
so 3 would be the same, it has two divisors
4, now 4 is gonna be different
it can divide by 1, 2 and 4 so it has three divisors, and hey!
This has more divisors than the others before it
so now this is the current winner
This is the highly composite number so far.
Let's see what 5 does, well 5 is a prime
so 5 loses
all right, way down there, 2.
So 4 is still the title holder?
Yeah, 4 is still winning,
but then 6 comes along
oh dear, 6, it can be divided by 1, 2, 3 and 6.
Four divisors, and now 6 is ahead of the game.
We keep going, oh 7, ah no 7 is no good, is a prime it has two divisors
8, does 8 do any better? So we can divide by 1, 2, 4 and 8
is has four divisors
and it's not better than 6 then
so four divisors, no it hasn't won anything, and 9 has three divisors so
it's 1, 3 and 9
10 has four divisors
11, prime, two divisors.
12 has six divisors, cause, I told you
12 is one of these numbers that have lots of things that divide into it
1, 2, 3, 4, 6 and 12. So that has six divisors,
12 is really good, really up there
and then so on.
So then yeah, well let's have a look at the title holders there, ok
highly composite numbers
let's write out the sequence
1, you're correct, 1 is there
2 is there
4, 6, 12
and if we carry on, 24, 36, 48, 60
60 is a good number, that's why we have 60 minutes in an hour, 60 seconds...
120, 180, 240
360's there like degrees in a circle
lots of things divide into 360, it's a good number to do
720 and 840 and so on. right
and they carry on like that.
So these are, this is our sequence of highly composite numbers
There's is a very important theorem in maths
called the Fundamental Theorem of Arithmetic
if you want its fancy name.
It means all positive whole numbers can be written as
a product of primes by multiplying prime numbers together
this is why primes are important, they're our building blocks for other numbers
they're our atoms for other numbers
So all other numbers can be written like this
if you had a number you'd have primes,
let's just call them just call them prime 1 to a power,
prime 2 to some power, prime 3 to some power,
and that would go on and then and you would end at some point you have the last prime here prime K to some
power
let's do an example so if you had the number 30 it's built up from primes it's
2 times 3 times 5. Primes that divide 30 has to be either 2 3 or 5. 7 doesn't
divide
yeah so if i had if i do another example had 550 it be 2 times 5 squared times 11
up three doesn't divide into it, seven doesn't divide into it
19 cannot divide into this number
I know if you want to do you want to have a factors you just use that idea
repeatedly
so the factors are the primes that divide into it repeatedly so I could
divide this by 25, 'cause I can divide by 5 and 5 again or if I want to divide by 10, I can divide it by 2 and then divide by 5.
So the factors are just all the possible combinations or permutations of these
prime atoms that you've been given to work with
They would all look like this all the factors would look like this would be
the same primes to be a P1 and P2
P3 to Pk you would have powers here, let's call them B1 and B2 and B3
Just these powers would be 0 or 1 or 2 or 3 or up to and including
the final thing
So anything less than what I've called a pair
Now those are your factors so how many factors are there just to show you how
many factors that are then the factors of a number of, let's call it n, are the
divisors of n is a well how many choices do I get for these powers and
it's this
it could be 0, 1, 2, 3, 4, anything
look to A1 so it be at A1 plus 1 multiplied by how many choices for the second prime power
0, 1, 2, 3, 4, anything up to A2 that's 1 plus A2 choices
You just do the same thing for each of these prime powers so that would go up
to the last one which is Ak plus 1 and that's how many divisorss the I'll do an
example of these shall I
So if I do like 30 and look for the divisors of 30 I can use this here formula
look called the powers are just one
How many choices would I have for each of these powers there's two choices there two
for that one two for that one
So all the powers are one and this is going to be 2 times 2 times 2 which is 8
and there are 8 divisors of 30 and if I did it for 550 slightly different 'cause I've
got this square in it so many choices here for the first prime power it is 2
and so I would have 12 factors of 550
so we've seen how to work out divisors and we should just check 5040 mentally
great 5040 let's see how many devices that has if you break it down into
primes then it's going to look like this
It's 2 to the power 4 multiplied by 3 squared multiplied by 5 multiplied by 7 and so let's look at the divisor formula
so we want divisors of 5040 and we can use the powers to help us work out
So it's just gonna be 5 by 3 by 2 by 2 and that's going to be 60 so there are 60
devices are 5040 which is greater than the number smaller and 5040 that makes
it highly composite a hundred years ago Ramanujan's those studying these and
notice three properties that highly composite numbers have to have which
I'll show you now and the not too difficult to understand but the first
property is the primes of the factorization of our highly composite
number have to be consecutive primes
I mean look that's what happened here you've got 5040 at the where they were
consecutive primes they were 2, 3, 5, and 7 if you look at 550 just
to compare it to something that doesn't work and that wasn't consecutive primes
and that was 2 by 5 squared by 11
now I know that this is not a highly composite number because I could replace
this 11 for one of the missing primes
I could replace it for 7 and it would be a smaller number but with the
same number of divisors same number of factors so it's not going to be highly
composite or the best choice would be if I picked a number which had consecutive
primes
so if I picked i'm going to use the same powers they like they are there so I'm
going to use this 2 by 3 squared by 5
yeah that's better that has consecutive primes it's a lot smaller number is 90
and 90 we can see has the same number of devices than 550
so this is much better choice than 550 so it failed
so yes if you've got a highly composite number of primes are consecutive that's the
that's kinda nice
the second thing Ramanujan noticed is the powers in their prime factorization they
have to be weakly decreasing they have to be going down like this
so you can see it here in 540 look 4, 2, 1, and 1
so they're going weakly down so they're not increasing but it didn't
happen here with my 550 didn't happen here with my 90 either
I tell you why because I can make a better choice
if I swapped the powers around and if i put them in a decreasing order it would
look like this
I can have a 2 squared multiplied by 3 multiplied by 5 so it's the same powers but in a decreasing order
that makes the number smaller as to why is that a 60
so now you've got 60 there has the same number of divisors and hey this is the
most better choice than 19 in fact 60 there's a highly composite number at the
third thing
Ramanujan noticed was these highly composite numbers all end with a with
with 1 as the last power so they always end up with a single prime there at the end
and so it doesn't end with a square and then there's actually a couple of
exceptions for that there are two exceptions
these are highly composite numbers that break that rule 4 is a highly
composite number
and that's 2 squared and the other number is a 36 which is a 2 squared by 3 squared
what Ramanujan showed is that highly composite numbers have to end with a
prime that has a power one or two and two has these two exceptions there and
everything else they all end with one with their prime power at the end
oh that's less obvious less obvious than hear the facts i showed you i took some
while to prove but it's another necessary condition for a highly
composite number
[Brady Haran] In addition to our usual supporters we'd like to thank audible.com for supporting
this video if you have to cover a few miles in planes or trains or automobiles
one of the best ways to pass the time is listening to audiobooks
so while you watch me driving here from Bristol to Nottingham let me tell you
about audible.com they've got a huge range of titles a great app and a good
offer for new customers but before I tell you about that first recommendation
and I'm going to suggest airframe by michael crichton it deals with the topic
i always find fascinating that is air crash investigations
maybe not want to listen to on a plane if you're a nervous flyer but definitely
worth your time
now audible are offering a free 30-day trial of their service which includes your
first book if you go to audible.com/numberphile using that URL will mean
they know you came from here that address again audible.com/numberphile
for the free trial I use audible
I think they're well worth a look and I'd like to thank them again for
supporting this numberphile video
[Brady Haran] Not very anti-prime [James Grime] Stop trying to make anti-prime a thing
Stop trying to make fetch a thing