# 5040 and other Anti-Prime Numbers - Numberphile

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• Published on Jul 6, 2016
• Audible: www.audible.com/numberphile (free trial)
Dr James Grime discusses highly composite numbers.
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• Science & Technology

## Comments • 2 860

• Wizard of the Valley 4 hours ago

i remember trying to find numbers that would divide by 1-10 in highschool. i did math some times when i was board

• Tuschedz 3 days ago

I don't see what is particularly useful about 5040. 2520 is half the size, and the only two factors less than 100 they don't share are 16 and 48.

• ecchi squid 3 days ago

So Plato wanted his citizens divided...got it.

• Adam Kelly 4 days ago

Why does the divisor formula add one to each factor?

• Kris Oluich 8 days ago

I never knew that Plato was a mathematician. Learn something new everyday.

• Cipri 11 days ago

So y!=anti-prime if y is natural

• deet0109 16 days ago +1

Zero is the most composite number. Every number goes into it.

• JayTemple 17 days ago

A key part of the Fundamental Theorem of Arithmetic is that the set of primes involved is unique except for order. By contrast, try factoring them over the numbers of the form a + bi, where a and be are integers and i is the square root of -1. Then 5 can be factored (1 + 2i)(1 - 2i), but it can also be factored (2 + i) (2 - i), not unique.

• Ryan Lynch 18 days ago

13:35 he watched mean girls!

• Red Orb 18 days ago

all the digits of these numbers add up to 3, 6, or 9.. and eventually they will only add up to 9... Don't know what that means but it reminds me of something Tesla said..

• What about 5040^5040?
Wouldn't that have 60^60 factors?

• Vishal Farma 21 day ago

What if you drew a graph and plotted all the pound of all the "current champions". Would you get a special line?

• Epsilon Karamazov 22 days ago

I don't get it.

• dmakk767 28 days ago

Anti-Prime for life!

• InsolubleToaster Month ago

So what makes the superabundants different from these? note that 5040 is the 19th in the sequence of both, after which they diverge.

• David H Braun Month ago

It's too bad television was called "television" so many years ago. Now it's too late for a shorter name, "TV" for example, to catch on. ;-) I think "Anti-prime" will catch on. It's not too late, and it's easier to call to mind and rolls off the lips easier than "highly composite number", which is a more descriptive name.

• Novel Poem Month ago

Fun Fact:
5039- Prime
5040- Anti Prime
5041- Square

I wonder if there is any other combination of three consecutive numbers with these properties!

• Pretzel Month ago

Isn’t 0 the biggest highly composite number because anything times zero is zero?

• Jason Îdo Feldman Month ago

ṼXV 5,040

• Alex Dixon Month ago

G64! would be a nice antiprime, since it’s divisible by every number leading up to G64

• lawrence wei Month ago

Even though 2 is prime, isn't it highly composite?

• lawrence wei Month ago

360 was made the number of degrees because it was an anti-prime

• DavidFMayerPhD Month ago

5040 is 7 factorial, so it is obviously divisible by all integers less than or equal to 7, as well as all products of any subset of the integers less than or equal to 7.
7! = 2*3*4*5*6*7
is obviously divisible by any similar product with one or more of the factors deleted:
2*3*4*5*6 pentuples
2*3*4*5*7
2*3*4*6*7
...............
2*3*4*7
2*5*6*7
4*5*6*7
......
triples
doubles
singles

• Anthony Nork Month ago

Is the last power always have to be 1 thing because the next prime is lower than the last prime squared?

• Yasmine Grant Month ago

Lol at the Mean Girls reference. I think I learned something 😊

• Logan Quinn 2 months ago

2^25 x 3^24 x 5^23 x 7^22 x 11^21 x 13^20 x 17^19 x 19^18 x 23^17 x 29^16 x 31^15 x 37^14 x 41^13 x 43^12 x 47^11 x 53^10 x 59^9 x 61^8 x 67^7 x 71^6 x 73^5 x 79^4 x 83^3 x 89^2 x 97 = 8.43842472722e+388 = d(325) :P

• Evan Fields 2 months ago

I read airframe. An action thriller plus a highly technical aviation lesson. Interesting book.

• LARAUJO 2 months ago +1

4 and 36 are exceptions because they're square numbers, so all their prime factors must be square as well.

• Jeremy Hoeven van der 2 months ago

8!

• Candy XXV 2 months ago

Wanna know a better anti-prime? 10080

• Davi Gurgel 2 months ago

Its very interesting to point that every single anti-prime is either a multiple or divisor of 12. Put on my list of reasons of why 12 is my third favorite number

• Davi Gurgel 2 months ago

If highly divisible numbers are "anti-prime" and two is highly divisible, than is two a anti-prime prime?

• Some Random Dragon 2 months ago

12:35 Ghost driver!

• Blue 913 2 months ago

12080..
Should this be 2 times more effective?
Should it have 120 factors of more?
Or less of course

• Mars 2 months ago +1

Something cool.... 7! = 5040

• D K 2 months ago

numbaaahh

• Fausto Moraes 2 months ago

Great video! Now I can explain why a complete circunference has 360 degrees and why an hour has 60 minutes. But... I have a request. Would you please make a video about the demonstration of the number of divisors "formula"? Thanks so much!

• Sturgeon 2 months ago

4:38 You meant to say that the fundamental theorem states that every positive whole number can be written as a UNIQUE product of primes. The word unique is super important here.

• Tore Braathen 2 months ago

LEET

• Chris Mahoney 2 months ago

I wonder if there's a prime number with highly composite numbers on either side

• Ethan Silverman 2 months ago

162,570,240

• Michael Tieber 2 months ago

I really wonder why Plato got 5040 instead of 2520...
I have the feeling he was looking for a number which is dividable by all the number up to ten. 2520 is the smallest number which fulfils this. So why he got 5040? It has those extra twelve devisors: 16, 48, 80, 112, 114, 240, 336, 560, 720, 1008, 1680, 5040 which one of this could be so important that 2520 is not the optimal number?

• Mars 2 months ago

Michael Tieber Well... 7! = 5040 you see.

• Levi NoPants 2.0 2 months ago

The video is 13:37

• Alm1t3 Cabbage 2 months ago

I just love how excited this guy gets about maths lol

• 7r0113r 3 months ago

when you start doing meth and mess with prime numbers for 3 days straight

• Patrickhh 3 months ago

13:37
Both numbers are prime
I see

• real gamer 3 months ago

Well, there‘s technically only one anti-prime: 0. It can be divided by all the numbers bigger than it, that‘s an infinite number of divisors.

• AntnPantnMantn 3 months ago

wouldn't it be nice if we counted in base 12

• Kenneth Schechter 3 months ago

He says numba instead of number

• AmountStax 3 months ago

Anti-prime is a much better name.

• nvshd 3 months ago

Should have mentioned this is why 12 is such a common number. Clocks, dozens, months, even the zodiac and the reason why eleven and twelve don't end in -teen (because ancient people commonly used base-12 counting instead of the base-10 we use today).

• Kai Sander 3 months ago

is this the reason behind tv resolution (960*2 by 1080)?

• Walt F. 3 months ago

Are all the HCNs (except the first few) divisible by 10?

• Zekrom Dravice 3 months ago

You can multiply 4and 36 by1 to prove his third theory right you can do it in any number and it won't change anything

• Elijah Neveski 3 months ago

Factorials are great anti primes

• Azzedine Usman 3 months ago

2^16*3^12*5^8*7^4*11^2*13*17*19*23 can you please check this number out if this is enough anti prime??

• I am _______ 3 months ago

2 is the lowest antiprime

• Long Plays 3 months ago

5040 = 7!, 5040 - 1 = 5039 is a prime number, and 5040 + 1 = 5041 is a perfect square, specifically 71^2. 71 is a prime.

• Kun Kodiik 3 months ago

5793445238736255798985527240000000 has 3628800 factors.

• Nazik Adam 3 months ago

No 125 :/

• Larry 3 months ago

Why not just do x! And x -> infinity. Doesn't that make it have all the factors? Does it work that way?

• Ashwin K V 4 months ago

Can anyone tell me what Ramanujan didn't predict in math? He's made India proud

• Yaddlezap 4 months ago

2,12,360,226800,523908000,15732957240000,...

• Andrew Hall 4 months ago

I wrote a program that finds the number with the highest amount of divisors under 1000000. The result was 720720 with 240 divisors.

• Edvinas Sungur 4 months ago

Plato... platonic solids.. mysterium cosmographium... vsauce...

• Matthew Writer 4 months ago

(2^3)(3^3)(5^3)(7^3)(11^3)(13^3)(101)=

• Pablo Pickaxo 4 months ago

How is nobody worried about the footage towards the end? The guy is clearly riding on the wrong side of the road!

• Jacob Opstad 4 months ago

I like the name "anti-prime"

• DV8 2 LECM 5 months ago

Is there a lower-bound composite that is made up of all natural numbers before it? Is that even possible?

• pavka14 5 months ago

The special numbers-
0-Can be divided by every number, yet is NOT an anti-prime, and is even
1-The only number with one factor, it divides by 1 and itself, yet is an anti-prime number
2-The only even prime number, yet is anti-prime
3-Made from 1 and 2, and is a prime
4-The first number with three factors, is anti-prime
5-A prime
11111110000000000111111111111000000000011111000000000000000000000000000001111110000000011111111
11111110000000000111111111111000000000011111000000000000000000000000000001111110000000011111111
11111110000000000111111111111000000000011111000000000000000000000000000001111110000000011111111
11111110000000000000000000000000000000011111111111111000000000000111111111111110000000011111111
11111110000000000000000000000000000000011111111111111000000000000111111111111110000000011111111
11111110000000000000000000000000000000011111111111111000000000000111111111111110000000011111111
11111110000000000000000000000000000000011111111111111000000000000111111111111110000000011111111
11111110000000000111111111111000000000011111111111111000000000000111111111111110000000011111111
11111110000000000111111111111000000000011111111111111000000000000111111111111111111111111111111
11111110000000000111111111111000000000011111000000000000000000000000000001111110000000011111111
11111110000000000111111111111000000000011111000000000000000000000000000001111110000000011111111
11111110000000000111111111111000000000011111000000000000000000000000000001111110000000011111111

• Jason Îdo Feldman 5 months ago

(7I+I)(7I-I)=7!
...5040...

• Holly Lamovsky 5 months ago

5040

• Filip Morawiec 5 months ago

That's because of
10!=5040

• CraftQueenJr 5 months ago

Can 2040 be accurately written with a ¡

• William Cook Miller 5 months ago

Was reading Plato's Laws, decided to look up the number 5040 on the internets, this video popped up in search. I love you guys.

• Rohan Sharma 5 months ago

Many of the numbers if you add each digit separately will give you a single digit sum of 9. Like 5040, 5+0+4+0 = 9, or 45360 4+5+3+6+0 = 18 = 1+8 = 9.

• Tommy Chan 5 months ago +1

Would be great if you talked about superior highly composite numbers

• Kepler 68 6 months ago

Under the definition they used there is only one highly composite number: 0 with infinite factors

• Lorenzo Sarria 6 months ago

The numbers are not anti prime
2 would be prime and anti prime

• Yenji Jem 6 months ago

Wait so 2 is both prime and antiprime? O.o

• Mxke Antto 1 6 months ago

• Parasitic Angel 6 months ago

It wouldn't be great for a city. The number itself has alot of factors. But then one person dies, and you get a prime number.

• Casimir III 6 months ago +1

I really don't like math, but I really like your channel, a lot.

• drchpt 6 months ago

I watched this before going to bed and my dream actually had to do with this, a battle for the greatest "antiprime"

• Lyri Metacurl 6 months ago

720720 is cool too. That's got a f*** ton of factors.

• Ye but 1080 got more dividers than 5040

• Marcus Mitchell 6 months ago

So 1 is both prime and anti-prime

• Sinar Zaito 6 months ago

If he knew that 7! is 5040 this video wouldn't have existed.

• locustkllr 6 months ago

does that mean we can find very large highly composite numbers by starting a huge chain of sequential prime numbers with descending powers, starting with 2 to the nth ? N being some enormous power?

• Who am I? 6 months ago

In school, I learnt something about factor sums and the names for numbers with certain factor sums.

• Xx Blaineworld xX 6 months ago

Well, 504,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

• Perihelion Orbit 6 months ago

If 5040 is anti prime, wouldn't that make 1 a super prime?

• Ayan Chavan 6 months ago

36 36 36 36 36 36
36 36 36 36 36 36
36 36 36 36 36 36
36 36 36 36 36 36
36 36 36 36 36 36
36 36 36 36 36 36

• Eclipsum 6 months ago

7! (factorial) is equals to 5040. So nothing special regards the most divisions side.
If you don't know why 5040 can be divided by 1, 2, 3.. 7 is cause all of those numbers are multiplied together meaning you can cancel one of those numbers out. Like this:
1 * 2 * [3] * 4* 5 * 6 * 7 / [3] = 1 * 2 * 4 * 5 * 6 * 7 (see result is missing the 3). Basic maths. Brackets are just making it easier to see the 3's.

• Wylliam Judd 6 months ago

Are all numbers that follow those rules highly composite?

• Vader 7 months ago

So you can have 18 possible piston driven engine types? From long ago in my Automotive class I was told that if you can evenly divide 360 the divisor is the number of pistons in an engine is possible. The prime factor of 360 is (2^2)(3^2)(5). Using that equation we get 3*3*2 = 18. Though in reality a 360 piston driven engine probably is impractical. I love this channel.

• Хекфе Вол 7 months ago +1

Why not to make a video about special types of factorials? Double, triple, etc.
25!! = 25*23*21*19*17*15*13*11*9*7*5*3*1 = 7905853580625
25!!! = 25*22*19*16*13*10*7*4*1 = 608608000
25!!!! = 25*21*17*13*9*5*1 = 5221125
25!!!!! = 25*20*15*10*5 = 375000
25!!!!!! = 25*19*13*7*1 = 43225
25!!!!!!! = 25*18*11*4 = 19800
25!!!!!!!! = 25*17*9*1 = 3825
25!!!!!!!!! = 25*16*7 = 2800
25!!!!!!!!!! = 25*15*5 = 1875

• Хекфе Вол 7 months ago

There are different types of factorials.
7! = 5040.
Superfactorial of 7 is product of all factorials up to it: 1*2*6*24*120*720*5040 = 125411328000.
Hyperfactorial of 7 is a product of 7 superfactorials: 745453331864786829312000000.