Phile=Lover in Greek. Numberphile= Lover of Numbers, Audiophile= Lover of Sound (usually music), Bibliophile= Lover of Books and Paedophile= Lover of Children (though used these days almost exclusively to mean inappropriate love).
instead, you know what Highly Composite Numbers are, as there is no such things as Anti-Prime Numbers. This way, the video title is misleading, even more for someone who already knew what HCNs are. ;c
I suppose we can bestow the name 'anti-prime' as long as that is not already taken. They can have two names, why not? M-31 is also known as the Andromeda Galaxy...
Last year, at age 59, I was at the prime of my life. I am in the composite (anti-prime) of my life now. On my birthday, I will start another prime year.
Well that would be 2- Anti Prime 3- Prime 4- Square but that's boring. Next one is 4,5,6. A group that isn't completely obvious is, for example, 47 prime, 48 anti-prime, 49 square. I've taken a bit of time and ran through the highly composite numbers listed on the wiki page, and found that there is no square within a margin of 2 near a highly composite number above 5040 and below at least 720720, meaning no such trios exist there. It would be interesting to see if they exist beyond that, though.
8! also meets all 3 requirements listed for "anti-primes": it's factorization would be 2^7 x 3^2 x 5 x 7, consecutive primes with decreasing powers and ends with a power of 1. However, the list is correct, 8! is NOT a highly composite number. The thing is, the 3 requirements are simply properties that all "anti-primes" have, but it is NOT a definitive test, there are false-positives, such as this one. To weed out false-positives you have to consider different ways we can manipulate the powers+1, though I'm not sure there's a simple way of describing it. For example, if we calculate the number of factors of 8!, we get 8x3x2x2=96. But we can rearrange this calculation to get 96 a different way, for example we can split that 8 into a 4x2 and move the 2 to the end: 4x3x2x2x2=96. Now let's reverse engineer this into a prime factorization (decrease each number by 1 and use it as a power of consecutive primes) we get 2^3 x 3^2 x 5 x 7 x 11, which is the factorization of 27720, which is smaller than 8! but has the same number of factors (96). Now, doing this does not always make the number smaller, for example if we split that 4 we get 3x2x2x2x2x2=96, but applying this to a prime factorization gives us 60060. Basically it has to do with the way you rearrange the factors of the number of factors (in this case 96), can result in making the prime factorization larger or smaller depending on how long the factorization winds up vs how big their powers get. (ie when went from 8! to 27720 the new factorization introduced a factor of 11, but the reduction in 2's exponent removed a factor of 16, so the overall effect made it smaller. But doing it again in this case introduces a 13, while the changes in exponents only removed a factor of 2 and a factor of 3 (total factor of 6), so the overall effect was an increase).
Out of curiosity, and because I can only hope to be as smart or well-educated: do false positives still occur if 'anti-primes' are numbers with a number of factors greater than _or equal to_ the largest number of factors for numbers less than it, rather than strictly greater than? Is there even a way to check that?
The most enthusiastic mathematician I've ever met puts him to shame. We're fairly convinced the guy consumes a colossal amount of methamphetamines for breakfast every day.
no for people that use base 12, they count 1 2 3 4 5 6 7 8 9 X E. they arent actually the letters, but new numbers that were made. but the new numbers look like an X and an E, so when typing we use those. numberphile did a video on it on 12/12/12.
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.
I learned about highly composite numbers the hard way. In 4th grade we played a dice game where you're on a number N, you roll the number D, then the number of steps you move is the remainder of N/D. First one to 100 wins. I landed on 60, and it took me a few turns to realize I lost.
@@peterstangl8295 60 (which is a highly composite number) is exactly divisible by all the numbers on the die (1 through 6). Therefore, the remainder when 60 is divided by the die outcome would always be zero, and you'll be stuck forever on the position 60 since you will only be able to move 0 step irrespective of the die outcome.
to be fair I think James just knows a lot about him as well, Euler was messing with this stuff long before and came up with the Euler Phi Function which gives us that total number of divisors. I think Ramanujan just found more patterns in it like he describes
12 is incredibly useful for web design, where you might want very flexible column layouts. If you have a grid of 12 columns, you could lay out a website into 1, 2, 3, 4, 6 or 12 parts or any combination thereof very easily. Just a little non-math tidbit. :P
That's why 2 is my most favourite number; first prime number, the only number sharing both the traits of prime and anti-prime, and the only even prime number! Moreover, no number would show this pattern n+n=n*n=n^n!
Makes sense. For any number m, every nth factor multiplied by the nth to the last factor gives you m. But if m is a square, the square root is the only factor that is both the nth factor and the nth to the last factor. Making the total amount of factors odd.
Perfect structure in this. I love how he came out with something confusing, then broke it down with excellent examples that kept me asking questions until he had it fully explained. Perfect pacing and video structure.
Finding out what an anti-prime is made me click on the video in the first place. I occasionally check out Numberphile videos but I probably would have missed this one under another title.
It's just Imperial English. Counterclockwise is anticlockwise in the Commonwealth as well. They prefer it, it just sounds weird to Americans, who don't use it as much.
In High School Algebra, I always struggled with Prime Factorization. I just didn't get it. Later I flunked out of College Algebra one of the things I remember running up against was factoring and struggling again Now I'm going through college again, and learning Prime Factors, and I remembered this video, and Dr James Grime explaining the Fundamental Theorem of Arithmetic and it just clicked. Its as easy as adding or subtracting for me now. I wish someone had explained it that way to me 20 years ago. Thank you Dr. Grime and Numberphile for making Math fun, interesting, and accessible for all of us.
I'm so glad to see you guys made a video on this topic. In school I often considered the number 60 and how many factors it has, especially the first six consecutively. I wondered for quite some time about the properties of 60 and whether there are other numbers like it, but never knew there was a name for the phenomena. Thanks again, Numberphiles!
(6:00) Prime factorisation is useful when you want to find the smallest number that can divide into two numbers. For sports, they wanted a framerate that could evenly split into 50 and 60 Hertz. 50 = 2×5² and 60 = 2²×3×5 Therefore we need a number that can make the fabrications for these two numbers above. That is 2²×3×5² = 300. That's why sports is recorded in 300 fps so it can air internationally. It also allows for slowmotion too.
3 years later, but whatever. Relatedly, the sampling rate on CDs is 44100, or 2*2*3*3*5*5*7*7, making subsampling by any product of two of its factors exceptionally easy. It's not a highly composite number, but that's because they didn't want it to be that exactly, they wanted it to have many small factors.
I like Numberphile videos so much. I must confess that I don't understand it all, but every now and then, I get something that makes my day and life greater. Thanks guys.
I remember having to solve for and write out the answer to 100P20 and I spent an hour multiplying by hand and checking my work in the end it was correct, but I wondered how I could write the same number in less space, and I didn't know it at the time, but I wrote it in prime factorization format. I love how I can watch these videos and relate them to things I've done that I didn't know were special.
"A highly composite number, sometimes called an antiprime number, is a positive integer with more divisors than any smaller positive integer has. The term was coined by Ramanujan (1915). However, Jean-Pierre Kahane has suggested that the concept might have been known to Plato, who set 5040 as the ideal number of citizens in a city as 5040 has more divisors than any numbers less than it." this is from wikipedia and its official this is anti-prime
12 is really interesting. (1^3)(2^2)(3^1) All base values increasing in consecutive order. All exponents decreasing in consecutive order. Sum of all the digits is 12. 6 factors, which is also highly composite. We really should be using base 12.
12 has also become the most highly standardized number for equally dividing the octave. So, right there is another thing that what you're saying is applicable to.
+Vandreren Well, yes and no. 1 is arguably just as unique as the primes above it, being that it isn't a product of any positive integers that are between 1 and itself. Any integer x is only unique until we reach 2x and above. However, we only concern ourselves with what's below a given integer, when deciding whether or not to label an integer as "unique". There's nothing below 1; so, unlike the integers above 1, there's nothing to judge it from. Not exactly a fair game. However, if we DID consider all integers above any prime integer x, we could no longer simply say it is unique, but rather that it has MORE uniqueness than the primes below, and LESS uniqueness than the primes above. 1 is arguably not un-unique, but just the prime with the least amount of uniqueness.
I love this channel. I think in numbers -- it's so difficult to block out, I can't even go on a car trip without seeing something crazy in every license plate I see. These sorts of videos help me try to explain to my mom how I see the world.
The runtime of 13:37 is also special, not just because they are primes, but because 1337 is how Leetspeak writes the word "LEET" (1337) which means "elite".
as a computer nerd, I go for 45045 which has all odd divisors through 15. Its binary is sweet - 1010 1111 1111 0101. Then can just bit shift to pick up even divisors.
+ You_just Yes, that is, I didn't. I've since picked up on it, from looking at other comments. BTW, 1337 even looks a lot like "Leet" when you turn it upside-down!
And I clicked on this because I am a bell ringer, and we love us some 5040 action! (5040 is the number of changes in a peal of seven bells or less, where the goal is to ring all of the permutations of seven without repeating a row anywhere. And who said the bell ringing couldn't be mathematical?)
1 is the mother number. It represents the essence of being (which is naturally 100%), while 0 represents the essence of nothing. All numbers thereafter are birth'd through 1 and each other.
I just wrote a python script that makes a list of these most-divisible numbers up to any limit, and I thought I was a genius... But now I see it's been done already. Probably EVERY mathematical thought has already been done.
After reading comments on TH-cam for a while I was under the impression that the whole world had suddenly become stupid. However, the comments on this video have restored my faith in humanity. Thanks, guys/girls.
I used to hate math in school, but I can't get enough of these videos. I've seen the video on the quaternions, I would like to see one on the octonions and how much we currently understand them. 🤓
no but he means anything he was teaching would be made into fun easy-to-understand examples instead of the usual maths teacher with monotonous voice saying "that's just how it's done" and "you just have to learn it." when you ask questions :P
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
This is kinda true: because days of the week are cyclical, {M, T, W, R, F, Sa, Su} is equivalent to {T, W, R, F, Sa, Su, M} and therefore these aren't *really* unique. Truly unique arrangements are 7!/7, or generally (n-1)!
They're mostly artifacts of older cultures that used non-decimal counting systems (mesopotamian numerals, for example, used base 60 and the greeks got their first astronomical tables from them), but yes, the greeks and romans did specifically keep the using the nonnative system because they liked being able to divide them in so many ways.
Even with decimals, repeating ones are often a total pain when it comes to quick calculations. (Possible to get deviations adding or multiplying them back together.) So these types of values may be handier than you'd think in modern applications.
At last! A reason I can give to my friends on the other side of the Atlantic as to why 60 Hz is better! (actually, it matters if you're building synchronous motors...)
...a personal favorite from childhood was 55,440. (equals LCM of 1 thru 10, times 22) A current favorite is the tautonymic, easy-to-decompose-into-factors 360,360.
Here’s a more imaginative answer: 101 (prime) as the difference of two squares: 101 = 10² - i² Or in other words, the factors are 101 = (10 + i)(10 - i) Are these called “Gaussian integers”?
Seems to me that every “anti-prime” is a valid number of divisors for another anti-prime. (2 has 2 divisors, 6 has 4, 12 has 6, 60 has 12, 360 has 24, 1260 has 36, 2520 has 48, 5040 has 60, etc.)
In fact some of what he's discussing simplifies a bit if for your prime factorization you include the entire set of prime numbers and have zeroes as the powers of primes that aren't in the factorization. For example 12 = 2^2 * 3^1 * 5^0 * 7^0 * 9^0 * ... . Note that you still get exactly one unique prime factorization for every counting number, it's just that you are instead looking at infinite sequences of exponents most of which are zero. When you do this then the first two rules of highly composite numbers can be condensed to just say that the sequence of exponents must be weakly decreasing. For instance, if you are "missing a prime factor" then that means you have a zero as an exponent which is followed by a non-zero exponent which means the sequence isn't decreasing. Also notice that the formula to calculate the number of factors still works, since if the exponent is zero that translates into multiplying the formula by one for that factor.
Jabberwocky I'm not sure what you're getting at saying "a^0 = 1 which isn't prime". 2^3 = 8 isn't prime either but it is part of the prime factorization of 40 = 2^3 * 5^1 = 2^3 * 3^0 * 5^1 * 7^0 * 11^0 *....
Justin Right, it's similar to how a decimal expansion is the infinite sum of a sequence of numbers of the form k*10^(-n) where k is an integer from 0 to 9 and n is a counting number. (Except that numbers can have more than one decimal expansion such as 0.500... = 0.4999... while all counting numbers only have one unique prime factorization.)
[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...)
I occasionally ebay one or send them to Patreon patrons... follow Numberphile on Twitter/Facebook/Patreon etc and I would usually let people know there!
+BobSkiz1 He said "I occasionally ebay one or send them to Patreon patrons... follow Numberphile on Twitter/Facebook/Patreon etc and I would usually let people know there!"
In the older Counter-Strike games one of the choices for picking "Terror" was the 1337/Elite Crew. Kind of turned into a meme before memes were a thing.
The Babylonians after jumping into a time machine and having a look at us: "Ok cool, you kept the 360 degrees in a circle, that's good ... 24 hour day, very nice, very nice .... the hours themselves still have 60 minutes at 60 seconds each, I see, never change a good thing .... and as a base for your number system you ... you _what_?!" Yep, we totally blew it there. Probably lost all respect in that moment.
Yes... base ten :/ We have the fingers on our hands to thank for that. Base 12 would be much better. We could represent 1/6, 1/3, 1/4, 1/2 by 0.2, 0.4, 0.3, 0.6 respectively.... If only... ps. There were civilisations who used sexagesimal - base 60
+Davy Ker I usually count to twelve on my fingers. I count the sections or joints on the inside of four of my fingers and use my thumb to point out which section. If you use both hands you can count up to 144 using one hand as the twelves and one as the ones. Come to think of it, I think one of Brady's channels talked about this.
A million dollar challenge: find a highly composite number "n" such that: sigma(n) > ln(harmonic(n)) * exp(harmonic(n)) + harmonic(n) If such a number exists, it will disprove Riemann's hypothesis. On the other hand, if you can show that no such number exists, then the Riemann's hypothesis will be marked as "proved" and you will win a million dollars.
1) sigma(n) is the sum of the positive divisors of n. For example, the positive divisors of 12, are [1, 2, 3, 4, 6, 12], therefore sigma(12) = 28. 2) harmonic(n) is the nth-harmonic number, which is the sum of reciprocals of the first n natural numbers (1/1 + 1/2 + 1/3 + ... + 1/n). 3) ln(x) is the natural logarithm of x. 4) exp(x) is e^x, where e is 2.71828... Highly composite numbers have lots of divisors, therefore sigma(n) is at its maximum (this special group of highly composite numbers, are called "colossally abundant numbers". See: oeis.org/A004490 ). In 2001, Jeffrey Lagarias (building on the work of Grönwall from 1913), showed that the Riemann hypothesis is equivalent with the statement that I wrote in my original comment (see Lagarias' paper here: arxiv.org/pdf/math/0008177v2.pdf ). Just for illustration, when n=5040: a) sigma(5040) = 19344 b) ln(harmonic(5040)) * exp(harmonic(5040)) + harmonic(5040) = 19836.31... In this specific case, a < b. If anyone can show that this holds true for all the numbers n > 1, then the Riemann's hypothesis would be proved correct. On the other hand, if anyone can find a counter-example, the Riemann's hypothesis would be disproved (very unlikely).
Yes I think I figured it out. The number is 92934939291874748381929399485848388881829922828881209993884777775811002939948585766788289919293984857675848838929199193994998819992992948472810298485757488291919293847575673719393948885888281919199399192929394858675747382819191203050012947365810294858488289191776528593999108876632819298192938884757575748382919192929384857575838.
The primes in the prime factorisation of a highly composite number not only need to be consecutive but also start at 2, for else you can shift down and get a smaller number with the same number of divisors. E.g. 3^3 * 5^2 * 7 cannot be highly composite because 2^3 * 3^2 * 5 is smaller and has the same number of divisors.
Confession, I would not have clicked on the video if it said "Highly Composite Numbers" I wanted to know what the anti-prime was
same
I was waiting for some Da Vinci code thing, I'll be honest. plus, I wanted to see what a numberphile was >.
Phile=Lover in Greek. Numberphile= Lover of Numbers, Audiophile= Lover of Sound (usually music), Bibliophile= Lover of Books and Paedophile= Lover of Children (though used these days almost exclusively to mean inappropriate love).
instead, you know what Highly Composite Numbers are, as there is no such things as Anti-Prime Numbers. This way, the video title is misleading, even more for someone who already knew what HCNs are. ;c
I suppose we can bestow the name 'anti-prime' as long as that is not already taken. They can have two names, why not? M-31 is also known as the Andromeda Galaxy...
Last year, at age 59, I was at the prime of my life. I am in the composite (anti-prime) of my life now. On my birthday, I will start another prime year.
Child of age 2 is both prime and anti prime
friendofbeaver when I was 12, it was the same thing
You just took that chance
friendofbeaver Happy birthday!
@Number mathematics so that's why children are lovable but confusing
Fun Fact:
5039- Prime
5040- Anti Prime
5041- Square
I wonder if there is any other combination of three consecutive numbers with these properties!
Well that would be
2- Anti Prime
3- Prime
4- Square
but that's boring. Next one is 4,5,6. A group that isn't completely obvious is, for example, 47 prime, 48 anti-prime, 49 square.
I've taken a bit of time and ran through the highly composite numbers listed on the wiki page, and found that there is no square within a margin of 2 near a highly composite number above 5040 and below at least 720720, meaning no such trios exist there. It would be interesting to see if they exist beyond that, though.
Kind of weird how many coincidences there are in math
Any number that's one above an anti-prime has a pretty high chance of being prime.
EDIT: or one below
There's infinite trios
@@thefreekinscientist 2 is prime tho.
Yet *every* time I make a batch of 12 cookies, there's *always* 5 people to share with. Time to make 5040 cookies.
***** You're right! I meant 5 people total. My mistake.
You only need 60 (augmenting "12" with "5" ability, still highly composite)
ToadStar100 what if you eat 2 extras?
ToadStar100 there is going to be 5039 people then. Sorry it's the rules.
12×5=60, 60÷12=5, so make 5÷nnnnnn... cookies.
2 is both prime and anti prime.
SporeZy the only number with that characteristic, since most people don't count 1 as prime!
@@AlgaeGaming 1 is not prime
@@AlgaeGaming one is not a prime
@@SYFTV1 one used to be a prime soooooo I don't think this is an argument worth having. One is just not a useful prime
@@derekwheeler4299 You say it as if any of the entire maths was useful at all
Wikipedia "A highly composite number (or anti-prime)"
ITS OFFICIAL!!!!!!!!
Timfoolery
"Wikipedia official"
You know anyone can edit wikipedia right? 😂
LitAquah, and there are a lot of people who edit Wikipedia, so in the end, it 'converges' to fact
Can we get to anti-prime likes
5040=7! btw. Just if anyone was wondering why so many factors.
Well that explains it.
Nice observation. So are 2!=2, 3!=6, 4!=24, 5!=120, 6!=720. But 8!=40320 which is not highly compatible... (Not visible at 4:33)
8! also meets all 3 requirements listed for "anti-primes": it's factorization would be 2^7 x 3^2 x 5 x 7, consecutive primes with decreasing powers and ends with a power of 1.
However, the list is correct, 8! is NOT a highly composite number. The thing is, the 3 requirements are simply properties that all "anti-primes" have, but it is NOT a definitive test, there are false-positives, such as this one. To weed out false-positives you have to consider different ways we can manipulate the powers+1, though I'm not sure there's a simple way of describing it.
For example, if we calculate the number of factors of 8!, we get 8x3x2x2=96. But we can rearrange this calculation to get 96 a different way, for example we can split that 8 into a 4x2 and move the 2 to the end: 4x3x2x2x2=96. Now let's reverse engineer this into a prime factorization (decrease each number by 1 and use it as a power of consecutive primes) we get 2^3 x 3^2 x 5 x 7 x 11, which is the factorization of 27720, which is smaller than 8! but has the same number of factors (96).
Now, doing this does not always make the number smaller, for example if we split that 4 we get 3x2x2x2x2x2=96, but applying this to a prime factorization gives us 60060. Basically it has to do with the way you rearrange the factors of the number of factors (in this case 96), can result in making the prime factorization larger or smaller depending on how long the factorization winds up vs how big their powers get. (ie when went from 8! to 27720 the new factorization introduced a factor of 11, but the reduction in 2's exponent removed a factor of 16, so the overall effect made it smaller. But doing it again in this case introduces a 13, while the changes in exponents only removed a factor of 2 and a factor of 3 (total factor of 6), so the overall effect was an increase).
Yeah, my first reaction when mentioned numbers with many factors was that I thought "Wouldn't I just take a factorial for that?"
Out of curiosity, and because I can only hope to be as smart or well-educated: do false positives still occur if 'anti-primes' are numbers with a number of factors greater than _or equal to_ the largest number of factors for numbers less than it, rather than strictly greater than? Is there even a way to check that?
I love how 2, a literal PRIME number is also a highly COMPOSITE number.
for lack of competition. :)
also, there are infinite prim numbers. only one of them is even.
My thought exactly
It's such an odd number
The prefix 'Anti-' Makes every word more interesting!
Yeah like the word "semitic".
Anti-interesting?
... or anti-boring...😜
So... How about being meta... Let's talk about anti-words !
@@grabern
That's dark
James grime : probably the most enthusiastic mathematician alive
have you watched the kleinbottle videos?
Well if I had his job, I would be too!
Matt Parker. Known for the Parker Square.
did you see the klein bottle guy?
The most enthusiastic mathematician I've ever met puts him to shame. We're fairly convinced the guy consumes a colossal amount of methamphetamines for breakfast every day.
I bet they all look nice when written in base 12, too.
Michael how did i not know this!!
Michael How could it end in E if it is base 12
Michael 11 is B
no for people that use base 12, they count 1 2 3 4 5 6 7 8 9 X E. they arent actually the letters, but new numbers that were made. but the new numbers look like an X and an E, so when typing we use those. numberphile did a video on it on 12/12/12.
and 0 too
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.
Actually, it's not the smallest. Try 360,360.
@@reubenmanzo2054. 360,360 is not divisible by 16.
@@Pseudify I stand corrected.
I'm a fan of 27720 for a similar reason. It only gets 1-12, but it's smaller and has the same three numerals.
"The third thing that you may have noticed..."
You flatter me, thricely.
I learned about highly composite numbers the hard way. In 4th grade we played a dice game where you're on a number N, you roll the number D, then the number of steps you move is the remainder of N/D. First one to 100 wins. I landed on 60, and it took me a few turns to realize I lost.
The choice of die had doomed you to lose.
i don't get it
@@peterstangl8295 60 (which is a highly composite number) is exactly divisible by all the numbers on the die (1 through 6). Therefore, the remainder when 60 is divided by the die outcome would always be zero, and you'll be stuck forever on the position 60 since you will only be able to move 0 step irrespective of the die outcome.
Nillie
*number theorist has entered the chat*
🤣🤣😂👍🏼
I love watching this guy talk. His energy is infectious!
@@greatestgameofall That is incredibly unfortunate! 😢
@@greatestgameofall Why are you spreading wrong information?
@@manuupadhyay1944 what did they say?
Numberphile has taught me that if anything cool happened in math, Ramanujan had something to do with it
and Gauss and Euler!
and Riemann and Pascal!
to be fair I think James just knows a lot about him as well, Euler was messing with this stuff long before and came up with the Euler Phi Function which gives us that total number of divisors. I think Ramanujan just found more patterns in it like he describes
Ramanujan made vastly bigger contributions to mathematics than this interesting tidbit.
I was just thinking something similar.
I will refer to highly composite numbers as anti primes from now on
+aragonaut thank you :)
1 is not Prime.
So 2 is both prime and antiprime... Sounds a little contradictory, but I suppose '2 is both prime and highly composite' sounds equally strange
1 is the Mother Number
pretty much
12 is incredibly useful for web design, where you might want very flexible column layouts. If you have a grid of 12 columns, you could lay out a website into 1, 2, 3, 4, 6 or 12 parts or any combination thereof very easily.
Just a little non-math tidbit. :P
Nice man. Now that's the world of software engineer and web developer, programmer. Awesome
That's also why there are 12 inches to a foot
So 2 is prime, but also anti-prime...
nice!
14 min late :
Oh my...
hence it's bipolar
= mutual annihilation LOL
That's why 2 is my most favourite number; first prime number, the only number sharing both the traits of prime and anti-prime, and the only even prime number! Moreover, no number would show this pattern n+n=n*n=n^n!
1+1=1*1=1^1
The highest number that divides 100% of the numbers from 1 to itself.
Redswap Ummm
1+1=2
1*1=1
1^1=1
Whoops
Lyri Metacurl That number would be 1. The only number that divides evenly into 1 is 1.
=n!
The first thing i thought when he said its an antiprime is: 2 is the only prime and antiprime at the same time
Another quote about 2:
"All primes are odd. 2 is VERY odd."
5040=7! btw
You know one thing at least Jon Snow
And this is why 7 is a magical number in the wizardry , Harry.
I think most factorials of prime numbers are highly composite
You're a factorial, Harry
Or 10!/6!
Fun fact: if the number of divisors is odd, the number is a square!
Makes sense. For any number m, every nth factor multiplied by the nth to the last factor gives you m. But if m is a square, the square root is the only factor that is both the nth factor and the nth to the last factor. Making the total amount of factors odd.
@@Gadget622 well, everything makes sense now. Thank you.
It's the same thing
@Ar'Khan _ Khizarkhajul those are prime factors
Ar'Khan _ Khizarkhajul
525 has 12 unique factors.
Perfect structure in this. I love how he came out with something confusing, then broke it down with excellent examples that kept me asking questions until he had it fully explained. Perfect pacing and video structure.
You still titled the video anti-prime numbers haha love your sense of humor! :D
It's Brady idea, let him have it. I think it's a better term than Highly Composite Number.
Finding out what an anti-prime is made me click on the video in the first place. I occasionally check out Numberphile videos but I probably would have missed this one under another title.
It's just Imperial English. Counterclockwise is anticlockwise in the Commonwealth as well. They prefer it, it just sounds weird to Americans, who don't use it as much.
Ray Harper Well, the joke obviously went over your head. Or you didn't even watch the video.
its not going to catch on. stop trying to make it a thing
Video duration: leet
Holy shit
19 minutes late..
In the thumbnails it says 13:38
Only a Doctor in Mathematics can have this much swag
Damn I am too late
In High School Algebra, I always struggled with Prime Factorization. I just didn't get it. Later I flunked out of College Algebra one of the things I remember running up against was factoring and struggling again
Now I'm going through college again, and learning Prime Factors, and I remembered this video, and Dr James Grime explaining the Fundamental Theorem of Arithmetic and it just clicked. Its as easy as adding or subtracting for me now. I wish someone had explained it that way to me 20 years ago.
Thank you Dr. Grime and Numberphile for making Math fun, interesting, and accessible for all of us.
I'm so glad to see you guys made a video on this topic. In school I often considered the number 60 and how many factors it has, especially the first six consecutively. I wondered for quite some time about the properties of 60 and whether there are other numbers like it, but never knew there was a name for the phenomena. Thanks again, Numberphiles!
Cool! Let's all count in base 5040 for a much more practical everyday life! :D
***** Of course, why would you not? It's not that much!
*many
So we have to remember 5040 different symbols in order to do simple arithmetic?
We already do, with time and degrees and a 12 month year.
+Ott no, we use 10 symbols for all of them :)
(6:00) Prime factorisation is useful when you want to find the smallest number that can divide into two numbers. For sports, they wanted a framerate that could evenly split into 50 and 60 Hertz.
50 = 2×5² and 60 = 2²×3×5
Therefore we need a number that can make the fabrications for these two numbers above. That is 2²×3×5² = 300. That's why sports is recorded in 300 fps so it can air internationally. It also allows for slowmotion too.
LCM
3 years later, but whatever.
Relatedly, the sampling rate on CDs is 44100, or 2*2*3*3*5*5*7*7, making subsampling by any product of two of its factors exceptionally easy.
It's not a highly composite number, but that's because they didn't want it to be that exactly, they wanted it to have many small factors.
@@OhhCrapGuy And video seem to go with 48000, which is 2*2*2*2*2*2*2*3*5*5*5, if you really want to divide by 2.
This also means that if you want to support both 44100 Hz and 48000 Hz, you need 2^7 * 3^2 * 5^3 * 7^2, which is 7056000 Hz or 7056 kHz.
Highly composite numbers are some of my favorite numbers but I didn't know exactly what they were called til now. There's so many ways to split 'em!
+Brandon Shaffer they're cool'
antiprimes are great numbers
is there a way to easily factor large numbers? like for example 20 digits up....?
conejo093 no, but 20 digits is nothing for your computer ;)
factoring is one of the milennia problems btw.
6 is still the best
I like Numberphile videos so much.
I must confess that I don't understand it all, but every now and then, I get something that makes my day and life greater.
Thanks guys.
I like how the first two properties sound interesting when you first hear them but have really obvious proofs. Anti-primes are nice to human brains.
I remember having to solve for and write out the answer to 100P20 and I spent an hour multiplying by hand and checking my work in the end it was correct, but I wondered how I could write the same number in less space, and I didn't know it at the time, but I wrote it in prime factorization format. I love how I can watch these videos and relate them to things I've done that I didn't know were special.
"A highly composite number, sometimes called an antiprime number, is a positive integer with more divisors than any smaller positive integer has. The term was coined by Ramanujan (1915). However, Jean-Pierre Kahane has suggested that the concept might have been known to Plato, who set 5040 as the ideal number of citizens in a city as 5040 has more divisors than any numbers less than it."
this is from wikipedia
and its official this is anti-prime
But do we know when the line, "sometimes called an antiprime number" was added; before or after this video was released?
@@countOfHenneberg Just checked on internet archive, antiprime was not included in the wikipedia before this video
4:05 All highly composite numbers above 3, can be reduced to 3, 6, 9 if you sum their digits. 5+0+4+0=9.
12 is really interesting.
(1^3)(2^2)(3^1)
All base values increasing in consecutive order.
All exponents decreasing in consecutive order.
Sum of all the digits is 12.
6 factors, which is also highly composite.
We really should be using base 12.
Vladimir Melnik your factorisation is also palindromic! But 1^3 is redundant, you could have chosen any power of 1 so it's not very special.
I would like a base 12 system
count me in
12 has also become the most highly standardized number for equally dividing the octave. So, right there is another thing that what you're saying is applicable to.
+Vandreren Well, yes and no. 1 is arguably just as unique as the primes above it, being that it isn't a product of any positive integers that are between 1 and itself. Any integer x is only unique until we reach 2x and above. However, we only concern ourselves with what's below a given integer, when deciding whether or not to label an integer as "unique". There's nothing below 1; so, unlike the integers above 1, there's nothing to judge it from. Not exactly a fair game. However, if we DID consider all integers above any prime integer x, we could no longer simply say it is unique, but rather that it has MORE uniqueness than the primes below, and LESS uniqueness than the primes above. 1 is arguably not un-unique, but just the prime with the least amount of uniqueness.
I love this channel. I think in numbers -- it's so difficult to block out, I can't even go on a car trip without seeing something crazy in every license plate I see. These sorts of videos help me try to explain to my mom how I see the world.
The runtime of 13:37 is also special, not just because they are primes, but because 1337 is how Leetspeak writes the word "LEET" (1337) which means "elite".
OK, Boomer!
Did the phrase anti-prime come off the cuff? Surely it can't have, it's too perfect!
totally - as he described it I just thought that must be its name... others have thought so too, I have since discovered.
How else are perfect things made?
It's from a Star Trek episode. ;) j/k
So if I multiply a prime by an anti-prime will they annihilate?
Nope, just makes a new number. A bit anti-climactic, don't you think? XD
It resets the universe
More like anti-primactic
+Reydriel 2 is prime AND anti prine. 2x2=4 which is also antiprime. 1 is antiprime and 2 is. 2*1=2 which is prime so you can get both
I think if the prime is high enough (i.e. higher than the highest prime factor of the anti-prime) you will get a new anti-prime.
as a computer nerd, I go for 45045 which has all odd divisors through 15. Its binary is sweet - 1010 1111 1111 0101. Then can just bit shift to pick up even divisors.
Can we talk about the length of the video?
I checked to see if it was highly composite or a prime or a perfect square before seeing it mentioned in another comment.
Karl Muster ha
+ You_just:
What, the 13:37, you mean? That's sexagesimal for 13·60 + 37 = 817 = 19·43.
What *did* you want to say about it?
ffggddss obviously you don't understand the meme
+ You_just
Yes, that is, I didn't.
I've since picked up on it, from looking at other comments.
BTW, 1337 even looks a lot like "Leet" when you turn it upside-down!
James making a Mean Girls reference is something none of us knew we needed but that has nonetheless completed our existence.
Recursive definition of antiprimes:
1) A given antiprime has more divisors than the last antiprime.
2) The first antiprime is 2.
13:37
that's pretty 1337
One thousand three hundred thirty seven is my number
0m9 1h4t'5 4m42in9
Pianoss lol I was born 13/37 actually
WashiestDrop198 wait wha-
What is the significance of 1337????
And I clicked on this because I am a bell ringer, and we love us some 5040 action! (5040 is the number of changes in a peal of seven bells or less, where the goal is to ring all of the permutations of seven without repeating a row anywhere. And who said the bell ringing couldn't be mathematical?)
*7!* = 5040
sqrt(5041) = *71*
sqrt(7! + 1) = 7*10+1
7! + 1 = (1+7*10)^2
I love how two is both a prime and an anti prime
Wait, so 2 is both a Prime and an Anti-Prime? So ist annihilates itself?
Just noticed someone else was a lot earlier.
+Pahckle I know, I just like the name and made a joke out of it.
Highly composite and prime at the same time still sounds odd.
+Pahckle clearly antiprime is the superior term
+Jan Wanders oops, I just commented that too!
I still think 2520 is neat. It only has 59 factors, but it's the first number that all numbers 1 through 10 divide into.
The factor to number ratio is much higher with this number. This should be the winner.
so 1 should be the winner because the ratio is exactly 1:1?
1 is the mother number. It represents the essence of being (which is naturally 100%), while 0 represents the essence of nothing. All numbers thereafter are birth'd through 1 and each other.
+Tiln TheModerator 2 is also 1 to 1 with 2 factors
100% off topic here, but you made me think about the Intel i5 2520M
This guy is so excited to share the cool bits about numbers and it makes me happy.
There goes Brady again with the naming of things that already have names and insisting that people use them.... 😂💜💜
Wolfram Mathworld diverts "anticrime" to highly composite numbers - I was not the first to think it unfortunately.
Batman does anticrime
yes
anticrime numbers for the win
Antiprime is far catchier than "Highly Composite Number"
This is a *prime* contender for the most fascinating topic I have watched being explained on your channel in my taste.
I just wrote a python script that makes a list of these most-divisible numbers up to any limit, and I thought I was a genius... But now I see it's been done already. Probably EVERY mathematical thought has already been done.
Screen resolutions are often composed of those numbers apparently. Makes total sense actually. :)
If I'm ever feeling too smart I watch Numberphile videos.
the only guy in the world that talk about numbers so enthusiastic.
the video has 5040 views at the time of writing this... damn.
Should have taken a screenshot.
thank you for not saying "should of".
it's sad the you're the exception.
+GraveUypo Arguably, that could be written "should 've" so it's ok in conversation.
And the video is kinda leet-long
+Ross Parlette this is not Grammarphile
After reading comments on TH-cam for a while I was under the impression that the whole world had suddenly become stupid. However, the comments on this video have restored my faith in humanity.
Thanks, guys/girls.
Then stop lurking on the wrong side of TH-cam
When I saw 5040 in the title I thought to myself "Oh, he is going to talk about 7!".... But he didn't mention it once.
I used to hate math in school, but I can't get enough of these videos. I've seen the video on the quaternions, I would like to see one on the octonions and how much we currently understand them. 🤓
A teacher like you back in school and I would have been a mathematician today! Keep it up!! :D
This is not the stuff you learn in a maths degree.
It's less what he's teaching, and more the passion he has.
no but he means anything he was teaching would be made into fun easy-to-understand examples instead of the usual maths teacher with monotonous voice saying "that's just how it's done" and "you just have to learn it." when you ask questions :P
+stefanozurich Not unless you take number theory. And do math in your spare time.
Thank you sir
The largest anti-prime I found so far is 55440 with 120 factors and it is the 28th anti-prime I found
Looks like Brady did a Parker square of naming these numbers
1:42 "I don't think it's going to catch on..."
Haaaaave you met Brady?
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
I'm glad you mentioned it was the 'ancient Greek philosopher' Plato, otherwise I'd be confusing him with all the other renowned Platos around.
lets make antiprime a thing!
no.
Stop trying to make antiprime happen!
I actually do like anti-prime better than "Highly composite number", it's shorter and reuses the same word (just with a prefix) xD
+Noel Goetowski so you are anti antiprime?
Wolfram Mathworld already recognises "Antiprime" as a synonym of "Highly composite number", so to some it extent it already is "a thing"
You could write a secret message in a number by having the power to an ASCII code. 2^65*3^66*5^65 but you get very big numbers
There are 5040 ways in which you can arrange days of a week!
Ashish Jog Write all permutations
you just meant 7 factorial indirectly ;)
Yea because 7! ="5040
This is kinda true: because days of the week are cyclical, {M, T, W, R, F, Sa, Su} is equivalent to {T, W, R, F, Sa, Su, M} and therefore these aren't *really* unique. Truly unique arrangements are 7!/7, or generally (n-1)!
@@bigfoot722
:D
It seems that our clocks and calendar systems use quite a few of these anti-prime numbers. I'm guessing this is intentional.
Circles, electricity standards in U.S., probably some other things.
Definitely, they are very useful
They're mostly artifacts of older cultures that used non-decimal counting systems (mesopotamian numerals, for example, used base 60 and the greeks got their first astronomical tables from them), but yes, the greeks and romans did specifically keep the using the nonnative system because they liked being able to divide them in so many ways.
Even with decimals, repeating ones are often a total pain when it comes to quick calculations. (Possible to get deviations adding or multiplying them back together.) So these types of values may be handier than you'd think in modern applications.
At last! A reason I can give to my friends on the other side of the Atlantic as to why 60 Hz is better!
(actually, it matters if you're building synchronous motors...)
...a personal favorite from childhood was 55,440. (equals LCM of 1 thru 10, times 22) A current favorite is the tautonymic, easy-to-decompose-into-factors 360,360.
Didn't realize prime factorization would be something that I'd ever watch about on youtube XD
10:36 the number magically becomes 540
Five hundred forty
Was reading Plato's Laws, decided to look up the number 5040 on the internets, this video popped up in search. I love you guys.
Brady you are amazing ! love your enthusiasm for the word anti-prime.:-)
James is amazing.
4:51 Every time something gets called an “atom”, it turns out to be divisible into smaller components. So far this hasn’t happened with primes ...
3 = 1.5 * 2
Here’s a more imaginative answer: 101 (prime) as the difference of two squares:
101 = 10² - i²
Or in other words, the factors are
101 = (10 + i)(10 - i)
Are these called “Gaussian integers”?
In school, I learnt something about factor sums and the names for numbers with certain factor sums.
'Uploaded 7 sec ago' damn that's early. And I'm not even subbed :(
You should subscribe!
Not subbed.
Pffft
PFFFT
Subbed as in subtitle or subscribe?
+Rizky Andyno Ramadhan subcribed
Why would he not be subtitled?
subscribed...><
I'm not kinda a math person. Sorry! But he makes quality vids tho
"I don't think it's going to catch on" - Names the video anti-prime anyway.
Seems to me that every “anti-prime” is a valid number of divisors for another anti-prime.
(2 has 2 divisors, 6 has 4, 12 has 6, 60 has 12, 360 has 24, 1260 has 36, 2520 has 48, 5040 has 60, etc.)
5,040 is also 7!
(! means factorial.)
First time reading this I though, wow how excited for a wrong statement, then I read you meant factorial xD
sqrt(5041) = *71*
*7!* = 5040
sqrt(7! + 1) = 7*10+1
7! + 1 = (1+7*10)^2
@@Rudxain wow
Any askers?
yes hes in the movie called nobody
And here I just always called them "factor-licious" numbers.
Actually, the only highly composite number is 0, because that has infinite factors, because it obviously has itself, and any number times 0 is 0.
Brady’s response to 1:45 is “**** you I can call the video what I want” 😂
Not related, but interesting is that the conversion factor for miles to feet, 5280, is equal to 2^5 x 3 x 5 x 11.
this video has really made me understand the world better, things like why are there 60 seconds in a minute etc. can all be explained by antiprimes
In fact some of what he's discussing simplifies a bit if for your prime factorization you include the entire set of prime numbers and have zeroes as the powers of primes that aren't in the factorization. For example 12 = 2^2 * 3^1 * 5^0 * 7^0 * 9^0 * ... . Note that you still get exactly one unique prime factorization for every counting number, it's just that you are instead looking at infinite sequences of exponents most of which are zero.
When you do this then the first two rules of highly composite numbers can be condensed to just say that the sequence of exponents must be weakly decreasing. For instance, if you are "missing a prime factor" then that means you have a zero as an exponent which is followed by a non-zero exponent which means the sequence isn't decreasing.
Also notice that the formula to calculate the number of factors still works, since if the exponent is zero that translates into multiplying the formula by one for that factor.
infinite sequences are mathematically difficult to deal with.
Lot's of problems become easier when you use infinite series, just look at Fourier analysis.
Good idea, this way every integer can be unambiguously defined only by the sequence of its exponents
Jabberwocky I'm not sure what you're getting at saying "a^0 = 1 which isn't prime". 2^3 = 8 isn't prime either but it is part of the prime factorization of 40 = 2^3 * 5^1 = 2^3 * 3^0 * 5^1 * 7^0 * 11^0 *....
Justin Right, it's similar to how a decimal expansion is the infinite sum of a sequence of numbers of the form k*10^(-n) where k is an integer from 0 to 9 and n is a counting number. (Except that numbers can have more than one decimal expansion such as 0.500... = 0.4999... while all counting numbers only have one unique prime factorization.)
lol the runtime is 13:37 I see you
this video was used in the wikipedia article on highly composite numbers! it is in the sources!
Maybe do an episode on 1337, huh?
What is special about it?
It is the length of this episode. Besides another fact...
the video length is 13:37
not a prime. 1337 = 7 x 191. you should be ashamed.
+thststth
Skaries never claimed it was a prime...
[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...)
Did nobody notice anything peculiar about the sequence of highly composite numbers?
12, 24, 36, 48,
60, 120, 180, 240,
360, 720,
840, 1260, 1680,
2520, 5040, 7560, 10080,
15120, 20160, 25200,
27720,
45360, 50400
Differences keep reappearing... Fascinating!
Where can I buy used Numberphile brown paper?
I occasionally ebay one or send them to Patreon patrons... follow Numberphile on Twitter/Facebook/Patreon etc and I would usually let people know there!
+Numberphile Thanks!
Strange. I can't see Numberphile's reply. Can someone enlighten me?
+BobSkiz1 He said "I occasionally ebay one or send them to Patreon patrons... follow Numberphile on Twitter/Facebook/Patreon etc and I would usually let people know there!"
TheIchigo1324 TY :)
video length is 13:37
dank
Elite!!!
What is leet and why is this video dank?
1337 pwnz0rz
In the older Counter-Strike games one of the choices for picking "Terror" was the 1337/Elite Crew. Kind of turned into a meme before memes were a thing.
Leet is where you replace letters with numbers, e.g. you might say "n00b" instead of "noob". If you try to replate 'leet' with numbers, you get 1337.
Many of the numbers if you add each digit separately will give you a single digit sum of 9. Like 5040, 5+0+4+0 = 9, or 45360 4+5+3+6+0 = 18 = 1+8 = 9.
The Babylonians after jumping into a time machine and having a look at us: "Ok cool, you kept the 360 degrees in a circle, that's good ... 24 hour day, very nice, very nice .... the hours themselves still have 60 minutes at 60 seconds each, I see, never change a good thing .... and as a base for your number system you ... you _what_?!" Yep, we totally blew it there. Probably lost all respect in that moment.
So let's use Pi instead as the base of our number system. ;-)
Yes... base ten :/ We have the fingers on our hands to thank for that. Base 12 would be much better. We could represent 1/6, 1/3, 1/4, 1/2 by 0.2, 0.4, 0.3, 0.6 respectively.... If only...
ps. There were civilisations who used sexagesimal - base 60
Davy Ker Yeah and you know who that civilization was? The Babylonians. Boom! ;)
+Davy Ker I usually count to twelve on my fingers. I count the sections or joints on the inside of four of my fingers and use my thumb to point out which section. If you use both hands you can count up to 144 using one hand as the twelves and one as the ones.
Come to think of it, I think one of Brady's channels talked about this.
Malachiore Point is, you're a couple millennia late with all those tricks.
A million dollar challenge: find a highly composite number "n" such that:
sigma(n) > ln(harmonic(n)) * exp(harmonic(n)) + harmonic(n)
If such a number exists, it will disprove Riemann's hypothesis. On the other hand, if you can show that no such number exists, then the Riemann's hypothesis will be marked as "proved" and you will win a million dollars.
Daniel Șuteu please explain the challenge. I don't understand the operations
1) sigma(n) is the sum of the positive divisors of n. For example, the positive divisors of 12, are [1, 2, 3, 4, 6, 12], therefore sigma(12) = 28.
2) harmonic(n) is the nth-harmonic number, which is the sum of reciprocals of the first n natural numbers (1/1 + 1/2 + 1/3 + ... + 1/n).
3) ln(x) is the natural logarithm of x.
4) exp(x) is e^x, where e is 2.71828...
Highly composite numbers have lots of divisors, therefore sigma(n) is at its maximum (this special group of highly composite numbers, are called "colossally abundant numbers". See: oeis.org/A004490 ). In 2001, Jeffrey Lagarias (building on the work of Grönwall from 1913), showed that the Riemann hypothesis is equivalent with the statement that I wrote in my original comment (see Lagarias' paper here: arxiv.org/pdf/math/0008177v2.pdf ).
Just for illustration, when n=5040:
a) sigma(5040) = 19344
b) ln(harmonic(5040)) * exp(harmonic(5040)) + harmonic(5040) = 19836.31...
In this specific case, a < b. If anyone can show that this holds true for all the numbers n > 1, then the Riemann's hypothesis would be proved correct. On the other hand, if anyone can find a counter-example, the Riemann's hypothesis would be disproved (very unlikely).
Yes I think I figured it out. The number is 92934939291874748381929399485848388881829922828881209993884777775811002939948585766788289919293984857675848838929199193994998819992992948472810298485757488291919293847575673719393948885888281919199399192929394858675747382819191203050012947365810294858488289191776528593999108876632819298192938884757575748382919192929384857575838.
1?
It is... Not 99999999999999999999999999999999999999999p99999
Or 5000000000
Or 200000000600000000
Prime number are more fundamental than what we are thinking about them
The primes in the prime factorisation of a highly composite number not only need to be consecutive but also start at 2, for else you can shift down and get a smaller number with the same number of divisors. E.g. 3^3 * 5^2 * 7 cannot be highly composite because 2^3 * 3^2 * 5 is smaller and has the same number of divisors.