# 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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Memphis Furman2 days ago110100100

Person Man Man5 days ago*Anti prime numbers ACTIVATE!!!!!!!!!*

Zach K5 days agoI think I made a highly composite number with π:

216π² = 2³ * 3³ * π² = 2131.8345506353014616682500559732

48 factors: 1, 2, 3, π (3.14), 4, 6, 2π (6.28), 8, 9, 3π (9.42), π² (9.87), 12, 4π (12.57), 18, 6π (18.85), 2π² (19.74), 24, 8π (25.13), 27, 9π (28.27), 3π² (29.61), 36, 12π (37.7), 4π² (39.48), 54, 18π (56.55), 6π² (59.22), 72, 24π (75.4), 8π² (78.96), 27π (84.82), 9π² (88.83), 108, 36π (113.1), 12π² (118.44), 54π (169.65), 18π² (177.65), 216, 72π (226.19), 24π² (236.87), 27π² (266.48), 108π (339.29), 36π² (355.31), 54π² (532.96), 216π (678.58), 72π² (710.61), 108π² (1065.92), 216π² (2131.83).

(shhhhhhhhh I know π isn't technically prime. And yes, I did type this by hand.)

Luz6 days agothe first property seems to be formulated wrong, actually

see

550 = 2^1 x 3^0 x 5^2 x 7^0 x 9^0 x 11^1

so it's not about not being factorized by consecutive primes

it's about not having one of those primes exponentiated to 0

it would be interesting to test it with a factorization using also negative numbers ex. 2^1 x 3^2 x 5^-4

MisterTutor20107 days agoThis would be the opposite if a prime square. That type of number would have a prime factorization of two identical prime numbers and as such would be divisible by only 3 numbers, itself, 1, and a single prime number. It could be called a minimally composite number.

Examples: 4, 9, 25, 49, 121, etc.

Come to think of it, such numbers could be called semi-prime powers since they have both prime factorization of two prime numbers (semi-prime) and it's prime factorization is composed of identical prime numbers (prime power).

bill bellychik9 days agoTough Titmouse

goroth019 days agoMy favorite is 2520, because it's the smallest number divisible by all the numbers 1-10.

dodo11 days ago^{+1}but the number 2520 is still special, because its the smallest number that has all the factors from 0 to 10

may Ho12 days ago2520 is the smallest number can divide 1 to 10

Kanishk Malkan14 days ago36 isn't anti prime

its 2^2 * 3^2

2^3 *3(24) is less than 36

now what? someone answer if I'm wrong...

:)

Racon Vid16 days agoIsn’t an even better one 1,000,000! = 1*2*3*4*5*6*7*8*9*10....*999,999,999*1,000,000

Nihal Ar25 days agoFactorial of every number is highly composite.

hsiF A25 days ago^{+1}ANTIPRIME!!!!

Inder Rai26 days agoIs there any odd numbers on this list?

Inder Rai26 days agoYou wouldn't think so but has it been proven not to be

Jack Baxter29 days agoAntiprime is the best possible name. Rewrite the books

Coola TöntenMonth agoI love the anti-prime epithet!

Daniel MoolmanMonth agoThere should be a term for numbers where you can get all natural numbers smaller than it by adding up its factors, only using each one once.

Wai Shing TseungMonth agoAntiprimeNumbers is there are all:

1,2,4,6,12,24,36,60,120,240,360,600,720,1200,2400,3600,7200,…

Daniel RodriguezMonth agoI wish Dr. James Grime would have continued with a friendly explanation of what are Superior Highly Composite Numbers (a term also coined by Ramanujan) and HOW you GENERATE Superior Highly Composite Numbers. The definition and method of creation of SHCN is not as obvious as that for Highly Composite Numbers, specially for infant students of mathematics (like myself). It would be great to have Numberphile and Dr. Grime making a video about SHCNs! ☺️

PaulPower4Month agoI'm a fan of the highly composite number 720,720 as the smallest number divisible by everything from 1 to 16 (since it's 720 x 1001, and 1001 is 7 x 11 x 13 - with 720 taking care of all the other factors) - plus it also looks pretty neat with the repeated digits.

jamie cruzMonth ago36=2^2+3^2+1^1. Fixed it!

Matthew WMonth ago232792560 is the first number that goes into all numbers up to 20 (and 21 and 22 as well)

2329089562800 goes into all numbers up to 30

Vincent HaeppMonth agoThey look all like multiples of 12...

Anita JaveriMonth agoCongrats Ramanujan...

ShortcutMonth agoWhat would 5040 look like, if you represent it as a shape drawn on paper, would it be a triangle? a square? both?

Lenin's NoseMonth agoSo 2 is both a prime and an antiprime? A synprime???

JPPR1410Month agoMaybe a little late but, if i understood correctly the fundamental theorem of arithmetics wasnt compeletly said. The theorem says in fact that any non prime number greater than one can be expressed as product of distinct primes but more important is that product is UNIQUE!!!! Which in the video isnt made very clear, at least to me

Kirill429Month ago13:37 is the perfect, most elite length of video

Gwen Simmons2 months ago^{+1}Loved his ‘anti-prime’! 😂😃🤣👍🏾TFS! Happy Holidays to you both! 🎄❄️☃️🤘🏾💫

BerwynSigns2 months agoIt's 13:37 EST and the video is 13:37 long.

Prakash Adhikari2 months agoI actually remember when I was little I wondered what do you call composite numbers with lots of factors.

Jerreu2 months agobasically the third rule can not be apply since “at the end“ an infinite amount of primes to the power of 0 follow, but well, that basically comes down to how you explained it

Jer Berus2 months agoI worked a lot with prime factorization when I was doing my mathematics thesis about group theory (with a focus on dihedral groups). I did a lot of combinatorics without realizing that was what I was doing. I understand this topic.

Ashton Smith2 months ago5040 is 7!

John J2 months agoBeautiful!

Packi Kittipob2 months ago4:30 2016!!!!!(x10)

anya zarembski2 months agough once again i’m reminded how much i love numbers

John Hooyer2 months agoWhat if we counted in Base 5040?

John Hooyer2 months agoI clicked on this video so fast! (I'm somewhat obsessed with the number 5040)

Semicedevine2 months agoI had a feeling that he found 5040 just by multiplying whole numbers together from 1 all the way up to 10...

1*2*3*4*5*6*7 = 5040

close enough, HMMMMMM.....

Wizard of the Valley2 months agoi remember trying to find numbers that would divide by 1-10 in highschool. i did math some times when i was board

Tuschedz2 months 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 squid2 months agoSo Plato wanted his citizens divided...got it.

Adam Kelly2 months agoWhy does the divisor formula add one to each factor?

Kris Oluich2 months agoI never knew that Plato was a mathematician. Learn something new everyday.

Cipri2 months agoSo y!=anti-prime if y is natural

deet01093 months ago^{+1}Zero is the most composite number. Every number goes into it.

JayTemple3 months 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 Lynch3 months ago13:35 he watched mean girls!

Red Orb3 months 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 Randomness3 months agoWhat about 5040^5040?

Wouldn't that have 60^60 factors?

Vishal Farma3 months agoWhat if you drew a graph and plotted all the pound of all the "current champions". Would you get a special line?

Epsilon Karamazov3 months agoI don't get it.

dmakk7673 months agoAnti-Prime for life!

InsolubleToaster3 months 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 Braun3 months 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 Poem3 months agoFun Fact:

5039- Prime

5040- Anti Prime

5041- Square

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

Pretzel3 months agoIsn’t 0 the biggest highly composite number because anything times zero is zero?

Jason Îdo Feldman3 months agoṼXV 5,040

Alex Dixon4 months agoG64! would be a nice antiprime, since it’s divisible by every number leading up to G64

lawrence wei4 months agoEven though 2 is prime, isn't it highly composite?

lawrence wei4 months ago360 was made the number of degrees because it was an anti-prime

DavidFMayerPhD4 months 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 Nork4 months agoIs the last power always have to be 1 thing because the next prime is lower than the last prime squared?

Yasmine Grant4 months agoLol at the Mean Girls reference. I think I learned something 😊

Logan Quinn4 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 Jackep4 months ago13:37

Evan Fields4 months agoI read airframe. An action thriller plus a highly technical aviation lesson. Interesting book.

LARAUJO4 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 der4 months ago8!

CandyXXV4 months agoWanna know a better anti-prime? 10080

Davi Gurgel4 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 Gurgel4 months agoIf highly divisible numbers are "anti-prime" and two is highly divisible, than is two a anti-prime prime?

Some Random Dragon4 months ago12:35 Ghost driver!

Blue 9134 months ago12080..

Should this be 2 times more effective?

Should it have 120 factors of more?

Or less of course

Mars4 months ago^{+1}Something cool.... 7! = 5040

D K4 months agonumbaaahh

Miles Boeddinghaus4 months ago13:37

Fausto Moraes4 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!

Sturgeon4 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.

Pringles_Glaz5 months agoLEET

Chris Mahoney5 months agoI wonder if there's a prime number with highly composite numbers on either side

Ethan Silverman5 months ago162,570,240

Michael Tieber5 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?

Mars4 months agoMichael Tieber Well... 7! = 5040 you see.

LevicusX5 months agoThe video is 13:37

Alm1t3 Cabbage5 months agoI just love how excited this guy gets about maths lol

Patrickhh5 months ago13:37

Both numbers are prime

I see

real gamer5 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.

AntnPantnMantn5 months agowouldn't it be nice if we counted in base 12

Kenneth Schechter5 months agoHe says numba instead of number

AmountStax5 months agoAnti-prime is a much better name.

nvshd5 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).

KSR NAN5 months agois this the reason behind tv resolution (960*2 by 1080)?

Walt F.5 months agoAre all the HCNs (except the first few) divisible by 10?

Zekrom Dravice6 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 Neveski6 months agoFactorials are great anti primes

Azzedine Usman6 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 _______6 months ago2 is the lowest antiprime

Long Plays6 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 Kodiik6 months ago5793445238736255798985527240000000 has 3628800 factors.