# 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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Prime numbers (more videos): bit.ly/primevids

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Yasmine Grant10 hours agoLol at the Mean Girls reference. I think I learned something 😊

Logan Quinn6 days 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 Jackep7 days ago13:37

Evan Fields8 days agoI read airframe. An action thriller plus a highly technical aviation lesson. Interesting book.

LARAUJO8 days 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 der9 days ago8!

Candy XXV9 days agoWanna know a better anti-prime? 10080

Davi Gurgel10 days 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 Gurgel10 days agoIf highly divisible numbers are "anti-prime" and two is highly divisible, than is two a anti-prime prime?

Some Random Dragon10 days ago12:35 Ghost driver!

Blue 91311 days ago12080..

Should this be 2 times more effective?

Should it have 120 factors of more?

Or less of course

Mars12 days ago^{+1}Something cool.... 7! = 5040

D K13 days agonumbaaahh

Miles Boeddinghaus19 days ago13:37

Fausto Moraes20 days 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!

Sturgeon20 days 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 Braathen22 days agoLEET

Chris Mahoney25 days agoI wonder if there's a prime number with highly composite numbers on either side

Ethan Silverman27 days ago162,570,240

Michael Tieber29 days 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?

Mars12 days agoMichael Tieber Well... 7! = 5040 you see.

Levi NoPants 2.029 days agoThe video is 13:37

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

7r0113rMonth agowhen you start doing meth and mess with prime numbers for 3 days straight

PatrickhhMonth ago13:37

Both numbers are prime

I see

real gamerMonth 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.

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

Kenneth SchechterMonth agoHe says numba instead of number

AmountStaxMonth agoAnti-prime is a much better name.

nvshdMonth 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 SanderMonth agois this the reason behind tv resolution (960*2 by 1080)?

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

Zekrom DraviceMonth 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 NeveskiMonth agoFactorials are great anti primes

Azzedine UsmanMonth 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 _______Month ago2 is the lowest antiprime

Long PlaysMonth 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 Kodiik2 months ago5793445238736255798985527240000000 has 3628800 factors.

Nazik Adam2 months agoNo 125 :/

Larry2 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 V2 months agoCan anyone tell me what Ramanujan didn't predict in math? He's made India proud

Yaddlezap2 months ago2,12,360,226800,523908000,15732957240000,...

Andrew Hall2 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 Sungur2 months agoPlato... platonic solids.. mysterium cosmographium... vsauce...

Matthew Writer2 months ago(2^3)(3^3)(5^3)(7^3)(11^3)(13^3)(101)=

Pablo Pickaxo2 months agoHow is nobody worried about the footage towards the end? The guy is clearly riding on the wrong side of the road!

Jacob Opstad2 months agoI like the name "anti-prime"

DV8 2 LECM3 months agoIs there a lower-bound composite that is made up of all natural numbers before it? Is that even possible?

pavka143 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 Feldman3 months ago(7I+I)(7I-I)=7!

...5040...

Holly Lamovsky3 months ago5040

Filip Morawiec3 months agoThat's because of

10!=5040

CraftQueenJr3 months agoCan 2040 be accurately written with a ¡

William Cook Miller3 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 Sharma4 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 Chan4 months ago^{+1}Would be great if you talked about superior highly composite numbers

Kepler 684 months agoUnder the definition they used there is only one highly composite number: 0 with infinite factors

Lorenzo Sarria4 months agoThe numbers are not anti prime

2 would be prime and anti prime

Yenji Jem4 months agoWait so 2 is both prime and antiprime? O.o

Mxke Antto 14 months agoWhat about 10080?

Parasitic Angel4 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 III4 months ago^{+1}I really don't like math, but I really like your channel, a lot.

drchpt4 months agoI watched this before going to bed and my dream actually had to do with this, a battle for the greatest "antiprime"

Lyri Metacurl4 months ago720720 is cool too. That's got a f*** ton of factors.

CubesCubesCubes Captain4 months agoYe but 1080 got more dividers than 5040

CubesCubesCubes Captain4 months ago*10080

Marcus Mitchell4 months agoSo 1 is both prime and anti-prime

Sinar Zaito4 months agoIf he knew that 7! is 5040 this video wouldn't have existed.

locustkllr4 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?4 months agoIn school, I learnt something about factor sums and the names for numbers with certain factor sums.

Xx Blaineworld xX4 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 Orbit4 months agoIf 5040 is anti prime, wouldn't that make 1 a super prime?

Ayan Chavan4 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

Eclipsum5 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 Judd5 months agoAre all numbers that follow those rules highly composite?

Vader5 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 Вол5 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 Вол5 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.

Horror Wagen5 months ago1337

Lux Ferre5 months agoUnless I'm missing something, the highest composite number with 4 prime bases and the smallest possible powers would be (2^4)*(3^3)*(5^2)*7 = 75600 which has 120 factors.

Mr.Lightning5 months ago1 and 2 are technically the most compisite numbers

Because

1 can be divided by only one number 1 and and can be decided to a non decal once so a score of 1/1

Heidgraben895 months agoIt's all alchemy, then.

avinotion5 months agoAll n! are Anti-Prime then

A. N.5 months ago7! is 5040, so are factorials more likely to be composite numbers than say, a composite number squared? (Sorry that was a bit of a loaded question.)

Lucas Hatfield5 months agoI don’t know if it is a Highly Composite Number, but the number (152700^2 + 31) has 64 divisors.

♫♪Ludwig van Beethoven♪♫5 months agoInfinity is the true prime

Max Jacobi5 months ago2520 has consecutive divisors from 1 to 10. So why Plato wanted 5040 citizens instead of 2520?

Simon Sozzi5 months agoAnti-prime numbers are so full of divisor it's packed like pickles! ...instead of "packed like sardines." It'll catch on. Just watch.

Gary Dixon5 months agoI think we should switch to base 5040

FimerianOfficial5 months ago^{+1}That is because 5040 is 7!

Nillie3 months agoAll the smaller factorials are also highly composite numbers. Alas, 8! is _not_ highly composite.

Troy Mihelcich5 months agoFunny thing is, I heard a guy refer to these types of numbers as anti-prime... maybe the other guy in the video wasn't too late?

PEROLER5 months agoit is also 7!

ZimZam1315 months agointegers

yeremia frans5 months ago90.

Shreyas Sarangi5 months agoMy first thought on seeing the thumbnail was 7!

Isaac Miles-Watt5 months ago2520 is an nice one

Eliot _6 months agoIn this video i learned that 2 was less of a prime number than 1.

Tixta6 months ago2016 has a nice few.

Craft Ore PL6 months agoThere are no Highly Composite Numbers because of 0

Leonaяdo DiCapяio6 months agoYour accent is so British that my country has become a British colony

Douwe Huysmans6 months agoHere's a nice anti-prime: 100!

LOLFlyingPotatoes6 months agoDo you even know it is anti-prime? Is there a proof that all factorial numbers are anti-prime?

DanielR16 months agoWhen 4 is an ANTIPRIME