Name: 5040 and other Anti-Prime Numbers - Numberphile
Uploaded: 2020-03-27T18:02:17.000Z
Duration: 13 min 38 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

To-infinitive

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.

Adverbs of frequency

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.

3, 4, 5, 6, 7 divides, 8 divides, 9 divides, 10 divides

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.

Then you'd divide it up into units of 5040.

5040, lots of divisors, plus it has more divisors

We now call this a Highly Composite Number.

[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

Let's take a look at properties of Highly Composite 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

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

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

it can divide by 1, 2 and 4 so it has three divisors, and hey!

This has more divisors than the others before it

This is the highly composite number so far.

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

so four divisors, no it hasn't won anything, and 9 has three divisors so

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,

So then yeah, well let's have a look at the title holders there, ok

60 is a good number, that's why we have 60 minutes in an hour, 60 seconds...

lots of things divide into 360, it's a good number to do

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

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

So all other numbers can be written like this

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

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

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

I know if you want to do you want to have a factors you just use that idea

Subtitles ListPlay Video

5040 and other Anti-Prime Numbers - Numberphile

obvious

fascinating

ancient

kinda