5040 and other Anti-Prime Numbers - Numberphile

  • Published on Jul 6, 2016
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    Dr James Grime discusses highly composite numbers.
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  • Science & TechnologyScience & Technology

Comments • 2 958

  • Memphis Furman
    Memphis Furman 2 days ago


  • Person Man Man
    Person Man Man 5 days ago

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

  • Zach K
    Zach K 5 days 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 6 days 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

  • MisterTutor2010
    MisterTutor2010 7 days 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 9 days ago

    Tough Titmouse

  • goroth01
    goroth01 9 days ago

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

  • dodo
    dodo 11 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 Ho
    may Ho 12 days ago

    2520 is the smallest number can divide 1 to 10

  • Kanishk Malkan
    Kanishk Malkan 14 days 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...

  • Racon Vid
    Racon Vid 16 days 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 25 days ago

    Factorial of every number is highly composite.

  • hsiF A
    hsiF A 25 days ago +1


  • Inder Rai
    Inder Rai 26 days ago

    Is there any odd numbers on this list?

    • Inder Rai
      Inder Rai 26 days ago

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

  • Jack Baxter
    Jack Baxter 29 days ago

    Antiprime is the best possible name. Rewrite the books

  • Coola Tönten
    Coola Tönten Month ago

    I love the anti-prime epithet!

  • Daniel Moolman
    Daniel Moolman Month 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 Month ago

    AntiprimeNumbers is there are all:

  • Daniel Rodriguez
    Daniel Rodriguez Month 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 Month 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 Month ago

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

  • Matthew W
    Matthew W Month 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 Month ago

    They look all like multiples of 12...

  • Anita Javeri
    Anita Javeri Month ago

    Congrats Ramanujan...

  • Shortcut
    Shortcut Month ago

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

  • Lenin's Nose
    Lenin's Nose Month ago

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

  • JPPR1410
    JPPR1410 Month 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 Month ago

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

  • Gwen Simmons
    Gwen Simmons 2 months ago +1

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

  • BerwynSigns
    BerwynSigns 2 months ago

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

  • Prakash Adhikari
    Prakash Adhikari 2 months ago

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

  • Jerreu
    Jerreu 2 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 2 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 2 months ago

    5040 is 7!

  • John J
    John J 2 months ago


  • Packi Kittipob
    Packi Kittipob 2 months ago

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

  • anya zarembski
    anya zarembski 2 months ago

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

  • John Hooyer
    John Hooyer 2 months ago

    What if we counted in Base 5040?

  • John Hooyer
    John Hooyer 2 months ago

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

  • Semicedevine
    Semicedevine 2 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 2 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 2 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.

  • ecchi squid
    ecchi squid 2 months ago

    So Plato wanted his citizens divided...got it.

  • Adam Kelly
    Adam Kelly 2 months ago

    Why does the divisor formula add one to each factor?

  • Kris Oluich
    Kris Oluich 2 months ago

    I never knew that Plato was a mathematician. Learn something new everyday.

  • Cipri
    Cipri 2 months ago

    So y!=anti-prime if y is natural

  • deet0109
    deet0109 3 months ago +1

    Zero is the most composite number. Every number goes into it.

  • JayTemple
    JayTemple 3 months ago

    A 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 Lynch
    Ryan Lynch 3 months ago

    13:35 he watched mean girls!

  • Red Orb
    Red Orb 3 months ago

    all 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 Randomness
    Bluestone Randomness 3 months ago

    What about 5040^5040?
    Wouldn't that have 60^60 factors?

  • Vishal Farma
    Vishal Farma 3 months ago

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

  • Epsilon Karamazov
    Epsilon Karamazov 3 months ago

    I don't get it.

  • dmakk767
    dmakk767 3 months ago

    Anti-Prime for life!

  • InsolubleToaster
    InsolubleToaster 3 months ago

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

  • David H Braun
    David H Braun 3 months ago

    It'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 Poem
    Novel Poem 3 months ago

    Fun Fact:
    5039- Prime
    5040- Anti Prime
    5041- Square

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

  • Pretzel
    Pretzel 3 months ago

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

  • Jason Îdo Feldman
    Jason Îdo Feldman 3 months ago

    ṼXV 5,040

  • Alex Dixon
    Alex Dixon 4 months ago

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

  • lawrence wei
    lawrence wei 4 months ago

    Even though 2 is prime, isn't it highly composite?

  • lawrence wei
    lawrence wei 4 months ago

    360 was made the number of degrees because it was an anti-prime

  • DavidFMayerPhD
    DavidFMayerPhD 4 months ago

    5040 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 quadruples

  • Anthony Nork
    Anthony Nork 4 months ago

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

  • Yasmine Grant
    Yasmine Grant 4 months ago

    Lol at the Mean Girls reference. I think I learned something 😊

  • Logan Quinn
    Logan Quinn 4 months ago

    2^25 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

  • Evan Fields
    Evan Fields 4 months ago

    I read airframe. An action thriller plus a highly technical aviation lesson. Interesting book.

    LARAUJO 4 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 der
    Jeremy Hoeven van der 4 months ago


  • CandyXXV
    CandyXXV 4 months ago

    Wanna know a better anti-prime? 10080

  • Davi Gurgel
    Davi Gurgel 4 months ago

    Its 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 Gurgel
    Davi Gurgel 4 months ago

    If highly divisible numbers are "anti-prime" and two is highly divisible, than is two a anti-prime prime?

  • Some Random Dragon
    Some Random Dragon 4 months ago

    12:35 Ghost driver!

  • Blue 913
    Blue 913 4 months ago

    Should this be 2 times more effective?
    Should it have 120 factors of more?
    Or less of course

  • Mars
    Mars 4 months ago +1

    Something cool.... 7! = 5040

  • D K
    D K 4 months ago


  • Fausto Moraes
    Fausto Moraes 4 months ago

    Great 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!

  • Sturgeon
    Sturgeon 4 months ago

    4:38 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_Glaz
    Pringles_Glaz 5 months ago


  • Chris Mahoney
    Chris Mahoney 5 months ago

    I wonder if there's a prime number with highly composite numbers on either side

  • Ethan Silverman
    Ethan Silverman 5 months ago


  • Michael Tieber
    Michael Tieber 5 months ago

    I 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?

    • Mars
      Mars 4 months ago

      Michael Tieber Well... 7! = 5040 you see.

  • LevicusX
    LevicusX 5 months ago

    The video is 13:37

  • Alm1t3 Cabbage
    Alm1t3 Cabbage 5 months ago

    I just love how excited this guy gets about maths lol

  • Patrickhh
    Patrickhh 5 months ago

    Both numbers are prime
    I see

  • real gamer
    real gamer 5 months ago

    Well, 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.

  • AntnPantnMantn
    AntnPantnMantn 5 months ago

    wouldn't it be nice if we counted in base 12

  • Kenneth Schechter
    Kenneth Schechter 5 months ago

    He says numba instead of number

  • AmountStax
    AmountStax 5 months ago

    Anti-prime is a much better name.

  • nvshd
    nvshd 5 months ago

    Should 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 NAN 5 months ago

    is this the reason behind tv resolution (960*2 by 1080)?

  • Walt F.
    Walt F. 5 months ago

    Are all the HCNs (except the first few) divisible by 10?

  • Zekrom Dravice
    Zekrom Dravice 6 months ago

    You 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 Neveski
    Elijah Neveski 6 months ago

    Factorials are great anti primes

  • Azzedine Usman
    Azzedine Usman 6 months ago

    2^16*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 _______
    I am _______ 6 months ago

    2 is the lowest antiprime

  • Long Plays
    Long Plays 6 months ago

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

  • Kun Kodiik
    Kun Kodiik 6 months ago

    5793445238736255798985527240000000 has 3628800 factors.