# 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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• Science & Technology

• Tommaso Morandini 21 hour ago

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

• 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 5 days ago +6

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

• Video is 13:37 long, two prime numbers

• Angela Kindness 9 days ago

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

• Aadarsh Sah 12 days ago

Are all integer factorials highly composite?

• Josh Howard 16 days ago

You mean antiprimes?

• Bernie O'Connor 16 days ago

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

• Ben Jammin Schreiber 16 days ago

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

• Tasty Rainbro 17 days ago

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

• LaGuerre19 21 day ago

James Grime for PM

• LaGuerre19 21 day ago +1

[Billy Joel singing]
...James Grime,
Anti-prime:
*_WHAT ELSE DO I HAVE TO SAY_*
(we didn't start the maths.

• Jason Katsanis 21 day ago

Could you make a video about Superior Highly Composite numbers?

• Antal László 23 days ago

12.252.240

• Master Monster 25 days ago

Ain't factorials like that?

• Rupert Bruce 27 days ago

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

• Icterine Tech Month ago

2520

• ralfoide Month ago

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

• So like factorials?

• Nouche Month ago +1

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

• shahnaz mirza Month ago

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

• 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.
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 X Month ago

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

• jusin james Month ago

ah NOOMBAH

• Evan Hartshorn Month ago

We must make it catch on

• 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 Month ago

Put some shoes on james

• Max Jytyla Month ago

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

• 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 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 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 Month ago

Mark Zuckerberg

• Ice Cream Month ago

yoy

• Destinitu DSO Month ago

Nerds... Go outside

• 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 2 months ago

2520 is the smallest number divisible by 1-10

• Brandon Sweat 2 months ago

anti prime is now a thing

• aytin horasan 2 months ago

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

• 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 2 months ago

basically all multiples of 12

• 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 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 2 months ago

came here from the Haskell Rosetta code article

• Adrian Wong 2 months ago +1

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

• the.abhiram.r 2 months ago

Nyumbah

• Katakana! 2 months ago +2

That's Numberwang!

• Dimitris Kalafatis 3 months ago

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

• Mike Sutton 3 months ago

Awesome!

• 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 3 months ago

is it 5040 or soho

• 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 3 months ago

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

• William Hinson 3 months ago

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

• 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 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.

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

• TKB 3 months ago

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

• htmlguy88 2 months ago

no as that would imply n

• They are building blocks ,atoms ,for other numbers

• Josh Mackay 3 months ago

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

• QuarioQuario54321 3 months ago

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

• Memphis Furman 3 months ago

110100100

• Person Man Man 3 months ago

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

• 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 4 months ago

the 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

• htmlguy88 2 months ago

means the same thing Also 550 is a failure.

• 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 4 months ago

Tough Titmouse

• goroth01 4 months ago

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

• 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 4 months ago

2520 is the smallest number can divide 1 to 10

• 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 2 months ago

24 has 8 factors, 36 has 9

• 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 4 months ago

Factorial of every number is highly composite.

• hsiF A 4 months ago +1

ANTIPRIME!!!!

• Inder Rai 4 months ago

Is there any odd numbers on this list?

• Inder Rai 4 months ago

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

• Jack Baxter 4 months ago

Antiprime is the best possible name. Rewrite the books

• Coola Tönten 4 months ago

I love the anti-prime epithet!

• 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 4 months ago

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

• 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 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 5 months ago

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

• 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 5 months ago

They look all like multiples of 12...

• Anita Javeri 5 months ago

Congrats Ramanujan...

• 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 5 months ago

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

• 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 5 months ago

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

• Gwen Simmons 5 months ago +1

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

• BerwynSigns 5 months ago

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

• 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 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 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 5 months ago

5040 is 7!

• John J 6 months ago

Beautiful!

• Packi Kittipob 6 months ago

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

• anya zarembski 6 months ago

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

• John Hooyer 6 months ago

What if we counted in Base 5040?

• John Hooyer 6 months ago

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

• 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 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 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.