5040 and other Anti-Prime Numbers - Numberphile

  • 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
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  • Science & TechnologyScience & Technology

Comments • 2 862

  • Tommaso Morandini
    Tommaso Morandini 21 hour ago

    I suspect that all the results of a factorial are "anti-primes"

  • D Nickaroo
    D Nickaroo 5 days ago

    The 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 Andrew
    Heinzen Andrew 5 days ago +6

    So wait... 2 is prime but also anti-prime??????? I'm confused

  • David Gallego Álvarez

    Video is 13:37 long, two prime numbers

  • Angela Kindness
    Angela Kindness 9 days ago

    55 440 **is even better** ( lowest number that is divisible by all the first twelve numbers )

  • Aadarsh Sah
    Aadarsh Sah 12 days ago

    Are all integer factorials highly composite?

  • Josh Howard
    Josh Howard 16 days ago

    You mean antiprimes?

  • Bernie O'Connor
    Bernie O'Connor 16 days ago

    I don't care what other people think, I'm calling these anti-prime numbers.

  • Ben Jammin Schreiber
    Ben Jammin Schreiber 16 days ago

    Yes but is it as anti-prime as 1 million factorial?

  • Tasty Rainbro
    Tasty Rainbro 17 days ago

    That rubiks in da backgeound makes angry. I wanna unscramble it :(

  • LaGuerre19
    LaGuerre19 21 day ago

    James Grime for PM

  • LaGuerre19
    LaGuerre19 21 day ago +1

    [Billy Joel singing]
    ...James Grime,
    (we didn't start the maths.
    they were always adding
    as the world was maddening...)

  • Jason Katsanis
    Jason Katsanis 21 day ago

    Could you make a video about Superior Highly Composite numbers?

  • Antal László
    Antal László 23 days ago


  • Master Monster
    Master Monster 25 days ago

    Ain't factorials like that?

  • Rupert Bruce
    Rupert Bruce 27 days ago

    @4:30 lol. Time to go write a compression algorithm...

  • Icterine Tech
    Icterine Tech Month ago


  • ralfoide
    ralfoide Month ago

    One of these 1337 speak videos, judging by its length.

  • The Extremely cool Seal

    So like factorials?

  • Nouche
    Nouche Month ago +1

    Is there any anti-prime that’s bigger than 4 but is a power of 2?

  • shahnaz mirza
    shahnaz mirza Month ago

    Can anyone explain to me how 2 is anti-prime?

  • Undefind
    Undefind Month ago

    I 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.
    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 X
    modi X Month ago

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

  • jusin james
    jusin james Month ago

    ah NOOMBAH

  • Evan Hartshorn
    Evan Hartshorn Month ago

    We must make it catch on

  • Nicholas Hibshman
    Nicholas Hibshman Month ago

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

  • Llewelyn Williams
    Llewelyn Williams Month ago

    Put some shoes on james

  • Max Jytyla
    Max Jytyla Month ago

    I love watching this guy talk. His energy is infectious!

  • Lauren Doe
    Lauren Doe Month ago

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

    • Lauren Doe
      Lauren Doe Month 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-Rivera
      Angel Mendez-Rivera Month ago

      Lauren 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 Wijs
    Mees de Wijs Month ago

    Mark Zuckerberg

  • Ice Cream
    Ice Cream Month ago


  • Destinitu DSO
    Destinitu DSO Month ago

    Nerds... Go outside

  • John B
    John B Month ago

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

  • Justin Nehls
    Justin Nehls 2 months ago

    2520 is the smallest number divisible by 1-10

  • Brandon Sweat
    Brandon Sweat 2 months ago

    anti prime is now a thing

  • aytin horasan
    aytin horasan 2 months ago

    came from oceanix city project 😊👏
    Nice Vid.
    one prime's prime multipliers also sequenced as primes😃

  • Starry Sunrose
    Starry Sunrose 2 months ago

    lol i love how 2 is, in addition to being the only even prime, also the only antiprime prime

  • director ʇᴉɯǝɹʞ
    director ʇᴉɯǝɹʞ 2 months ago

    basically all multiples of 12

  • smanzoli
    smanzoli 2 months ago

    Have 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 C
    Louis C 2 months ago

    Just 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.

  • eldersprig
    eldersprig 2 months ago

    came here from the Haskell Rosetta code article

  • Adrian Wong
    Adrian Wong 2 months ago +1

    Highly Composite Numbers = Antiprime. It caught on. Read wikipedia :) and quora

  • the.abhiram.r
    the.abhiram.r 2 months ago


  • Katakana!
    Katakana! 2 months ago +2

    That's Numberwang!

  • Dimitris Kalafatis
    Dimitris Kalafatis 3 months ago

    Don’t you like how 2, a prime, is also an anti prime

  • Mike Sutton
    Mike Sutton 3 months ago


  • Connor Sponsler
    Connor Sponsler 3 months ago

    So 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 Tanasov
    Georgiy Tanasov 3 months ago

    is it 5040 or soho

  • dittbub
    dittbub 3 months ago

    what 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

  • Dozenazer
    Dozenazer 3 months ago

    4:32 That's what I love 2016 for!

  • William Hinson
    William Hinson 3 months ago

    So 2 is prime, but also a highly composite number???

  • Francis Luglio
    Francis Luglio 3 months ago

    I'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?

  • fredact
    fredact 3 months ago

    So 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 Official

    LOL, so smart and he can’t solve a Rubik’s Cube! Look behind him on the table.

  • TKB
    TKB 3 months ago

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

    • htmlguy88
      htmlguy88 2 months ago

      no as that would imply n

  • comments are more interesting than videos

    They are building blocks ,atoms ,for other numbers

  • Josh Mackay
    Josh Mackay 3 months ago

    Did you reference Mean Girls at the end??? hahahaha

  • QuarioQuario54321
    QuarioQuario54321 3 months ago

    The only highly composite number is 0, which has Absolute Infinity factors

  • Memphis Furman
    Memphis Furman 3 months ago


  • Person Man Man
    Person Man Man 3 months ago

    *Anti prime numbers ACTIVATE!!!!!!!!!*

  • Zach K
    Zach K 3 months ago

    I 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.)

  • Luz
    Luz 4 months ago

    the first property seems to be formulated wrong, actually
    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

    • htmlguy88
      htmlguy88 2 months ago

      means the same thing Also 550 is a failure.

  • MisterTutor2010
    MisterTutor2010 4 months ago

    This 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 bellychik
    bill bellychik 4 months ago

    Tough Titmouse

  • goroth01
    goroth01 4 months ago

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

  • dodo
    dodo 4 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 Ho
    may Ho 4 months ago

    2520 is the smallest number can divide 1 to 10

  • Kanishk
    Kanishk 4 months ago

    36 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...

    • htmlguy88
      htmlguy88 2 months ago

      24 has 8 factors, 36 has 9

  • Racon Vid
    Racon Vid 4 months ago

    Isn’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 Ar
    Nihal Ar 4 months ago

    Factorial of every number is highly composite.

  • hsiF A
    hsiF A 4 months ago +1


  • Inder Rai
    Inder Rai 4 months ago

    Is there any odd numbers on this list?

    • Inder Rai
      Inder Rai 4 months ago

      You wouldn't think so but has it been proven not to be

  • Jack Baxter
    Jack Baxter 4 months ago

    Antiprime is the best possible name. Rewrite the books

  • Coola Tönten
    Coola Tönten 4 months ago

    I love the anti-prime epithet!

  • Daniel Moolman
    Daniel Moolman 4 months ago

    There 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 Tseung
    Wai Shing Tseung 4 months ago

    AntiprimeNumbers is there are all:

  • Daniel Rodriguez
    Daniel Rodriguez 4 months ago

    I 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! ☺️

  • PaulPower4
    PaulPower4 5 months ago

    I'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 cruz
    jamie cruz 5 months ago

    36=2^2+3^2+1^1. Fixed it!

  • Matthew W
    Matthew W 5 months ago

    232792560 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 Haepp
    Vincent Haepp 5 months ago

    They look all like multiples of 12...

  • Anita Javeri
    Anita Javeri 5 months ago

    Congrats Ramanujan...

  • Shortcut
    Shortcut 5 months ago

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

  • Lyle Cohen
    Lyle Cohen 5 months ago

    So 2 is both a prime and an antiprime? A synprime???

  • JPPR1410
    JPPR1410 5 months ago

    Maybe 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

  • Kirill429
    Kirill429 5 months ago

    13:37 is the perfect, most elite length of video

  • Gwen Simmons
    Gwen Simmons 5 months ago +1

    Loved his ‘anti-prime’! 😂😃🤣👍🏾TFS! Happy Holidays to you both! 🎄❄️☃️🤘🏾💫

  • BerwynSigns
    BerwynSigns 5 months ago

    It's 13:37 EST and the video is 13:37 long.

  • Prakash Adhikari
    Prakash Adhikari 5 months ago

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

  • Jerreu
    Jerreu 5 months ago

    basically 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 Berus
    Jer Berus 5 months ago

    I 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 Smith
    Ashton Smith 5 months ago

    5040 is 7!

  • John J
    John J 6 months ago


  • Packi Kittipob
    Packi Kittipob 6 months ago

    4:30 2016!!!!!(x10)

  • anya zarembski
    anya zarembski 6 months ago

    ugh once again i’m reminded how much i love numbers

  • John Hooyer
    John Hooyer 6 months ago

    What if we counted in Base 5040?

  • John Hooyer
    John Hooyer 6 months ago

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

  • Semicedevine
    Semicedevine 6 months ago

    I 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 Valley
    Wizard of the Valley 6 months 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
    Tuschedz 6 months 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.