# 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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Wizard of the Valley4 hours agoi remember trying to find numbers that would divide by 1-10 in highschool. i did math some times when i was board

Tuschedz3 days agoI 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 squid3 days agoSo Plato wanted his citizens divided...got it.

Adam Kelly4 days agoWhy does the divisor formula add one to each factor?

Kris Oluich8 days agoI never knew that Plato was a mathematician. Learn something new everyday.

Cipri11 days agoSo y!=anti-prime if y is natural

deet010916 days ago^{+1}Zero is the most composite number. Every number goes into it.

JayTemple17 days agoA 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 Lynch18 days ago13:35 he watched mean girls!

Red Orb18 days agoall 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..

Bluestone Randomness21 day agoWhat about 5040^5040?

Wouldn't that have 60^60 factors?

Vishal Farma21 day agoWhat if you drew a graph and plotted all the pound of all the "current champions". Would you get a special line?

Epsilon Karamazov22 days agoI don't get it.

dmakk76728 days agoAnti-Prime for life!

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

David H BraunMonth agoIt'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 PoemMonth agoFun Fact:

5039- Prime

5040- Anti Prime

5041- Square

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

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

Jason Îdo FeldmanMonth agoṼXV 5,040

Alex DixonMonth agoG64! would be a nice antiprime, since it’s divisible by every number leading up to G64

lawrence weiMonth agoEven though 2 is prime, isn't it highly composite?

lawrence weiMonth ago360 was made the number of degrees because it was an anti-prime

DavidFMayerPhDMonth ago5040 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*5 quadruples

2*3*4*7

2*5*6*7

4*5*6*7

......

triples

doubles

singles

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

Yasmine GrantMonth agoLol at the Mean Girls reference. I think I learned something 😊

Logan Quinn2 months ago2^25 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

Jack Kinda Emerald Pickaxe Jackep2 months ago13:37

Evan Fields2 months agoI read airframe. An action thriller plus a highly technical aviation lesson. Interesting book.

LARAUJO2 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 der2 months ago8!

Candy XXV2 months agoWanna know a better anti-prime? 10080

Davi Gurgel2 months agoIts 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 Gurgel2 months agoIf highly divisible numbers are "anti-prime" and two is highly divisible, than is two a anti-prime prime?

Some Random Dragon2 months ago12:35 Ghost driver!

Blue 9132 months ago12080..

Should this be 2 times more effective?

Should it have 120 factors of more?

Or less of course

Mars2 months ago^{+1}Something cool.... 7! = 5040

D K2 months agonumbaaahh

Miles Boeddinghaus2 months ago13:37

Fausto Moraes2 months agoGreat 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!

Sturgeon2 months ago4:38 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 Braathen2 months agoLEET

Chris Mahoney2 months agoI wonder if there's a prime number with highly composite numbers on either side

Ethan Silverman2 months ago162,570,240

Michael Tieber2 months agoI 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?

Mars2 months agoMichael Tieber Well... 7! = 5040 you see.

Levi NoPants 2.02 months agoThe video is 13:37

Alm1t3 Cabbage2 months agoI just love how excited this guy gets about maths lol

7r0113r3 months agowhen you start doing meth and mess with prime numbers for 3 days straight

Patrickhh3 months ago13:37

Both numbers are prime

I see

real gamer3 months agoWell, 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.

AntnPantnMantn3 months agowouldn't it be nice if we counted in base 12

Kenneth Schechter3 months agoHe says numba instead of number

AmountStax3 months agoAnti-prime is a much better name.

nvshd3 months agoShould 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 Sander3 months agois this the reason behind tv resolution (960*2 by 1080)?

Walt F.3 months agoAre all the HCNs (except the first few) divisible by 10?

Zekrom Dravice3 months agoYou 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 Neveski3 months agoFactorials are great anti primes

Azzedine Usman3 months ago2^16*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 ago2 is the lowest antiprime

Long Plays3 months ago5040 = 7!, 5040 - 1 = 5039 is a prime number, and 5040 + 1 = 5041 is a perfect square, specifically 71^2. 71 is a prime.

Kun Kodiik3 months ago5793445238736255798985527240000000 has 3628800 factors.

Nazik Adam3 months agoNo 125 :/

Larry3 months agoWhy not just do x! And x -> infinity. Doesn't that make it have all the factors? Does it work that way?

Ashwin K V4 months agoCan anyone tell me what Ramanujan didn't predict in math? He's made India proud

Yaddlezap4 months ago2,12,360,226800,523908000,15732957240000,...

Andrew Hall4 months agoI wrote a program that finds the number with the highest amount of divisors under 1000000. The result was 720720 with 240 divisors.

Edvinas Sungur4 months agoPlato... platonic solids.. mysterium cosmographium... vsauce...

Matthew Writer4 months ago(2^3)(3^3)(5^3)(7^3)(11^3)(13^3)(101)=

Pablo Pickaxo4 months agoHow is nobody worried about the footage towards the end? The guy is clearly riding on the wrong side of the road!

Jacob Opstad4 months agoI like the name "anti-prime"

DV8 2 LECM5 months agoIs there a lower-bound composite that is made up of all natural numbers before it? Is that even possible?

pavka145 months agoThe 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 Feldman5 months ago(7I+I)(7I-I)=7!

...5040...

Holly Lamovsky5 months ago5040

Filip Morawiec5 months agoThat's because of

10!=5040

CraftQueenJr5 months agoCan 2040 be accurately written with a ¡

William Cook Miller5 months agoWas 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 Sharma5 months agoMany 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 Chan5 months ago^{+1}Would be great if you talked about superior highly composite numbers

Kepler 686 months agoUnder the definition they used there is only one highly composite number: 0 with infinite factors

Lorenzo Sarria6 months agoThe numbers are not anti prime

2 would be prime and anti prime

Yenji Jem6 months agoWait so 2 is both prime and antiprime? O.o

Mxke Antto 16 months agoWhat about 10080?

Parasitic Angel6 months agoIt 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 III6 months ago^{+1}I really don't like math, but I really like your channel, a lot.

drchpt6 months agoI watched this before going to bed and my dream actually had to do with this, a battle for the greatest "antiprime"

Lyri Metacurl6 months ago720720 is cool too. That's got a f*** ton of factors.

CubesCubesCubes Captain6 months agoYe but 1080 got more dividers than 5040

CubesCubesCubes Captain6 months ago*10080

Marcus Mitchell6 months agoSo 1 is both prime and anti-prime

Sinar Zaito6 months agoIf he knew that 7! is 5040 this video wouldn't have existed.

locustkllr6 months agodoes 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 agoIn school, I learnt something about factor sums and the names for numbers with certain factor sums.

Xx Blaineworld xX6 months agoWell, 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 Orbit6 months agoIf 5040 is anti prime, wouldn't that make 1 a super prime?

Ayan Chavan6 months ago36 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

Eclipsum6 months ago7! (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 Judd6 months agoAre all numbers that follow those rules highly composite?

Vader7 months agoSo 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 Вол7 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 Вол7 months agoThere 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.