# 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.

More links & stuff in full description below ↓↓↓

Continues and extra footage: thexvid.com/video/PF2GtiApF3E/video.html

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

www.antiprime.net

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Tommaso Morandini21 hour agoI suspect that all the results of a factorial are "anti-primes"

D Nickaroo5 days agoThe combined radii of the Earth and the Moon is 5040 miles.

This number is a property of the number 10. 7 is related to 10 in a special way.

5040 = 7! = 7 x 8 X 9 X 10. 10 is the only number with this property.

Heinzen Andrew5 days ago^{+6}So wait... 2 is prime but also anti-prime??????? I'm confused

David Gallego Álvarez8 days agoVideo is 13:37 long, two prime numbers

Angela Kindness9 days ago55 440 **is even better** ( lowest number that is divisible by all the first twelve numbers )

Aadarsh Sah12 days agoAre all integer factorials highly composite?

Josh Howard16 days agoYou mean antiprimes?

Bernie O'Connor16 days agoI don't care what other people think, I'm calling these anti-prime numbers.

Ben Jammin Schreiber16 days agoYes but is it as anti-prime as 1 million factorial?

Tasty Rainbro17 days agoThat rubiks in da backgeound makes angry. I wanna unscramble it :(

LaGuerre1921 day agoJames Grime for PM

LaGuerre1921 day ago^{+1}[Billy Joel singing]

...James Grime,

Anti-prime:

*_WHAT ELSE DO I HAVE TO SAY_*

(we didn't start the maths.

they were always adding

as the world was maddening...)

Jason Katsanis21 day agoCould you make a video about Superior Highly Composite numbers?

Antal László23 days ago12.252.240

Master Monster25 days agoAin't factorials like that?

Rupert Bruce27 days ago@4:30 lol. Time to go write a compression algorithm...

Icterine TechMonth ago2520

ralfoideMonth agoOne of these 1337 speak videos, judging by its length.

The Extremely cool SealMonth agoSo like factorials?

NoucheMonth ago^{+1}Is there any anti-prime that’s bigger than 4 but is a power of 2?

Nouche6 days ago^{+1}@ineedanamebut thisisntcapitalisedproperly : Basically it can be divided by smaller or equal to it powers of two.

ineedanamebut thisisntcapitalisedproperly6 days ago65536?

shahnaz mirzaMonth agoCan anyone explain to me how 2 is anti-prime?

UndefindMonth agoI prefer the squence: 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, 360360, 360360, 360360, 720720, ...

This sequence is also high divisibility. However, the focus is on divisibility of all integers 1 to n, not the highest total factors.

1÷1=1

2÷1=2, 2÷2=1

6÷1=6, 6÷2=3, 6÷3=2

12÷1,2,3,4 = 12,6,4,3

60÷1,2,3,4,5 = 60,30,20,15,10

60 repeats since 60÷6=10

1 to 5 gets us 1 to 6.

Generating the Sequence:

d(1) = 1.

For all n>1,

if n = p^x where p is prime then d(n) = p×d(n-1)

else, d(n) = d(n-1)

Every time we reach a new prime, we need that entirely new factor.

Every time we reach a prime to a power, we need one additional factor of that prime than ever before. So if either n=p or n=p^x, we need to multiple by p to get the next divisibility.

If n is not prime nor a prime to a power, then we already have any necessary factors to ensure divisibility. So we get to keep our old value.

modi XMonth agoI think this also means a highly composite number needs to start with 2 in the prime factorization

jusin jamesMonth agoah NOOMBAH

Evan HartshornMonth agoWe must make it catch on

Nicholas HibshmanMonth agoIs 5040 the lowest number divisible by all numbers 1-10? I can’t quite work it out easily

Llewelyn WilliamsMonth agoPut some shoes on james

Max JytylaMonth agoI love watching this guy talk. His energy is infectious!

Lauren DoeMonth agoI say that 4 is the most "anti-prime" number. It can be divided by 75% of all numbers at or below it.

Lauren DoeMonth ago@Angel Mendez-Rivera Yes, you are correct. I did realize this when I made my remark, but I came to the decision that an anti-prime could not be a prime number.

Angel Mendez-RiveraMonth agoLauren Doe You are wrong. 2 is more anti-prime than 4, since it is divisible by every number before it or at it. Yes, 2 is an anti-prime number AND a prime number, the only number with this property.

Mees de WijsMonth agoMark Zuckerberg

Ice CreamMonth agoyoy

Destinitu DSOMonth agoNerds... Go outside

John BMonth agoI was somewhat infuriated by "(a1+1)x(1+a2)" and even more infuriated when it was animated "(a1+1)x(a2+1)"

Justin Nehls2 months ago2520 is the smallest number divisible by 1-10

Brandon Sweat2 months agoanti prime is now a thing

aytin horasan2 months agocame from oceanix city project 😊👏

Nice Vid.

one prime's prime multipliers also sequenced as primes😃

Starry Sunrose2 months agolol i love how 2 is, in addition to being the only even prime, also the only antiprime prime

director ʇᴉɯǝɹʞ2 months agobasically all multiples of 12

smanzoli2 months agoHave tried numbers up to 100.000

Here are the ones with at least 100 divisors:

45360 100

50400 108

55440 120

60480 112

65520 120

69300 108

70560 108

71280 100

73920 112

75600 120

79200 108

80640 108

81900 108

83160 128

84240 100

85680 120

87360 112

88200 108

90720 120

92400 120

93600 108

95040 112

95760 120

97020 108

98280 128

99792 100

The very easy to understand code (for dummies) in Excel VBA (yes, it´ll freeze you Excel for some minutes):

Sub Macro_Count_Divisors()

Dim a, b, c, d As Integer

Dim x As Single

For a = 1 To 100000

Cells(a, 1).Value = a

For b = 1 To a

x = a / b

d = Int(a / b)

z = x - d

If z = 0 Then

c = c + 1

End If

Next b

Cells(a, 2).Value = c

c = 0

Next a

End Sub

Louis C2 months agoJust find the prime factorization and determine all possible permutations of the primes that divide it. I don't see much importance to this video in at the 3 minute mark.

eldersprig2 months agocame here from the Haskell Rosetta code article

Adrian Wong2 months ago^{+1}Highly Composite Numbers = Antiprime. It caught on. Read wikipedia :) and quora

the.abhiram.r2 months agoNyumbah

Katakana!2 months ago^{+2}That's Numberwang!

Dimitris Kalafatis3 months agoDon’t you like how 2, a prime, is also an anti prime

Mike Sutton3 months agoAwesome!

Connor Sponsler3 months agoSo 2 is both the first prime and the first anti-prime.

If you want to talk physics, 2's superposition should be the main article of discussion.

Georgiy Tanasov3 months agois it 5040 or soho

dittbub3 months agowhat about high composite numbers that exclude sub factors. eg for '12' you would include 6 and 4 but not 3 and 2 (since they are factors of 6 and 4). And then what about numbers that have the most primes as factors. i guess you could predict that one. 1x2, 1x2x3, 1x2x3x5, 1x2x3x5x7, etc

htmlguy882 months ago^{+1}aka primorials.

Dozenazer3 months ago4:32 That's what I love 2016 for!

William Hinson3 months agoSo 2 is prime, but also a highly composite number???

Francis Luglio3 months agoI've noticed something. 96 is not a highly composite number. But 96 is (2^5)x(3^1). Where as, the preceding HCNs are 48 (2^4)x(3^1) and 60 (2^2)x(3^1)x(5^1) and the following is 120 (2^3)x(3^1)x(5^1). Is there a way to predict whether the next HCN involves increasing a1 vs adding another prime factor? I guess take the derivative of the two d(n) formulas you'd then have?

fredact3 months agoSo all you metric system fanboys, note that 10 is not on the list. Our forefathers long ago figured out that 12, 24, 60 were much better uses for everyday units like measurement and time.

The Emerald Men Official3 months agoLOL, so smart and he can’t solve a Rubik’s Cube! Look behind him on the table.

TKB3 months agoAside from 1 and 2, is there a number divisible by all the numbers before it?

htmlguy882 months agono as that would imply n

comments are more interesting than videos3 months agoThey are building blocks ,atoms ,for other numbers

Josh Mackay3 months agoDid you reference Mean Girls at the end??? hahahaha

QuarioQuario543213 months agoThe only highly composite number is 0, which has Absolute Infinity factors

Memphis Furman3 months ago110100100

Person Man Man3 months ago*Anti prime numbers ACTIVATE!!!!!!!!!*

Zach K3 months 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.)

Luz4 months 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

htmlguy882 months agomeans the same thing Also 550 is a failure.

MisterTutor20104 months 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 bellychik4 months agoTough Titmouse

goroth014 months agoMy favorite is 2520, because it's the smallest number divisible by all the numbers 1-10.

dodo4 months ago^{+1}but the number 2520 is still special, because its the smallest number that has all the factors from 0 to 10

may Ho4 months ago2520 is the smallest number can divide 1 to 10

Kanishk4 months 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...

:)

htmlguy882 months ago24 has 8 factors, 36 has 9

Racon Vid4 months 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 Ar4 months agoFactorial of every number is highly composite.

hsiF A4 months ago^{+1}ANTIPRIME!!!!

Inder Rai4 months agoIs there any odd numbers on this list?

Inder Rai4 months agoYou wouldn't think so but has it been proven not to be

Jack Baxter4 months agoAntiprime is the best possible name. Rewrite the books

Coola Tönten4 months agoI love the anti-prime epithet!

Daniel Moolman4 months 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 Tseung4 months agoAntiprimeNumbers is there are all:

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

Daniel Rodriguez4 months 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! ☺️

PaulPower45 months 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 cruz5 months ago36=2^2+3^2+1^1. Fixed it!

Matthew W5 months 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 Haepp5 months agoThey look all like multiples of 12...

Anita Javeri5 months agoCongrats Ramanujan...

Shortcut5 months agoWhat would 5040 look like, if you represent it as a shape drawn on paper, would it be a triangle? a square? both?

Lyle Cohen5 months agoSo 2 is both a prime and an antiprime? A synprime???

JPPR14105 months 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

Kirill4295 months ago13:37 is the perfect, most elite length of video

Gwen Simmons5 months ago^{+1}Loved his ‘anti-prime’! 😂😃🤣👍🏾TFS! Happy Holidays to you both! 🎄❄️☃️🤘🏾💫

BerwynSigns5 months agoIt's 13:37 EST and the video is 13:37 long.

Prakash Adhikari5 months agoI actually remember when I was little I wondered what do you call composite numbers with lots of factors.

Jerreu5 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 Berus5 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 Smith5 months ago5040 is 7!

John J6 months agoBeautiful!

Packi Kittipob6 months ago4:30 2016!!!!!(x10)

anya zarembski6 months agough once again i’m reminded how much i love numbers

John Hooyer6 months agoWhat if we counted in Base 5040?

John Hooyer6 months agoI clicked on this video so fast! (I'm somewhat obsessed with the number 5040)

Semicedevine6 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 Valley6 months agoi remember trying to find numbers that would divide by 1-10 in highschool. i did math some times when i was board

Tuschedz6 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.