8128 and Perfect Numbers - Numberphile
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- Опубликовано: 18 сен 2024
- For many years, 8128 was the largest known perfect number. But what is a perfect number?
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This video features Dr James Grime. James' own RUclips channel full of maths stuff can be found at / singingbanana
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"One million seven hundred ninety one thousand..."
At this point I thought, well, that's not very big.
"...digits long."
Oh. I see.
Oh dayum
incremental players: pathetic
⁹⁸⁹⁶⁰⁷²⁶⁰⁴.....
Gggggggggh
Vvvvvvvvvvvvvv
Fr everyone gangsta until he says "digits long"
Also, every (known) perfect number is represented in binary as a number of digits (we'll say 'n') followed by a number of digits (n-1). That is, 6 = 110, 28 = 11100, 496 = 111110000, 8128 = 1111111000000 etc. Also, the reciprocals of all divisors of all (known) perfect numbers sum to 2. For example, 6: (1/1 + 1/2 + 1/3 + 1/6 = 2), 28: (1/1 + 1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 2), 496: (1/1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/31 + 1/62 + 1/124 + 1/248 + 1/496 = 2) etc. :)
In these cases, n is a prime number. n = 2, 3, 5, 7. Is that a consistent pattern or a coincidence.
@@silvervaliant Yes, this is a rule - you can prove that all perfect numbers are equal to 2^(p-1) * (2^p)-1 where (2^p)-1 is a Mersenne prime, i.e. p is prime. This formula when converted to binary will always give a number of 1s equal to p, followed by a number of 0s equal to p-1. Hope that makes sense!
33550336 in binary is 1111111111111000000000000 (n=13)
8589869056 in binary is 111111111111111110000000000000000 (n=17)
137438691328 in binary is 1111111111111111111000000000000000000 (n=19)
2305843008139952128 in binary is 1111111111111111111111111111111000000000000000000000000000000 (n=31)
and so on :)
@@silvervaliant I noticed that too. Based on this, it looks like there are indeed infinitely many perfect numbers and since in binary the last digit is always zero, it means perfect numbers are always even.
@@silvervaliant Ah it looks like n is not prime, but the reverse of a Mersenne prime. It went from 7 directly to 13, skipping 11.
Okay (:
James: The largest known perfect number is 1 million...
Me: hmm, that's kind of small..
James: digits long
Me: oh.
Yup
:-D
hehe, exactly
noneedforid same reaction
@Avinash Sangale
No no
No k ko
6 is the most perfect number, since not only 1+2+3=6, but 1x2x3=6
wow.
nicholas giles
_oh wait_
3!
@Владимир Иванович Данилко But 4 uses copys of the same factor
Владимир Иванович Данилко you’ve reused those numbers...
I think it's a pity that there was a huge missed opportunity in this video: They could have mentioned that there's a one-to-one relationship between even perfect numbers and Mersenne primes. This was first conjectured by Euclid and finally proven by Euler two thousand years later.
_That_ I find even _more_ interesting than just the perfect numbers themselves.
I think you, sir, would like their more recent video with the topic of Mersenne primes
Actually, we know that all even perfect numbers can be generated by plugging a Mersenne prime into a particular expression. We still haven't proven if odd perfect numbers exist, and if we determine the existence of one, it will disprove the notion of a one-to-one relationship between Mersenne primes and perfect numbers.
We have only proven that every Mersenne prime comes with a perfect number, we have not proven that there are no OTHER perfect numbers out there. I do believe there are limiting proofs out there (if I'm not mistaken, it's proven that there are no even perfect numbers that are NOT related to a Mersenne prime), but we don't have proof of a one-to-one relationship.
They've done a video on that too...
Edit: Though to be honest they don't explain why.
⁹⁸⁹⁶⁰⁷²⁶⁰⁴.....
Would love to show the ancient Greeks a number almost two million digits long :)
You: *sniffs a maths book*
The timeline:
They don't even know what a digit is
Arabic Numerals don’t exist yet, you could not show them in our digits that we use.
The largest known perfect number seems to be 34,850,340 digits long (found in February 2013), according to Wikipedia. The 1,791,864 digit-long number was the largest known as of 1997. When the video was uploaded, the largest was 25,956,377 digits long.
would 6 be even more perfect? since its factors (discluding 6) also multiply together to make six, 3 x 2 x 1 (or 3!) = 6
Gracious Carrot +Retro he is talking about perfect numbers... can the other ones do that... 28 doesn't as well as the rest of perfect numbers.
its usually the ones with 2 factors other than 1, that can do that not all factors. also adding them does not give same number. which makes 6 a special case in both categories
ILoveAnime actually there is a way to calculate the number that does this, just take a prime and multiply it by 2
eragontapper14 it works with every two primes multiplied with each other.
I know right :')
+Retro I think he was talking about all the proper factors E.G. 12=(1,2,3,4,6,12) 1*2*3*4*6=144
now, in 2021, the largest known perfect number has 49724095 digits
I like how the video that is about perfect numbers is exactly 4:20 long.
True
You're right man
Nice
That’s so funny. I laughed. Just like this: haha.
I am watching this in 2024 and there are 51 perfect numbers currently
53*
I like the fact that if you take the largest factor of a perfect number and divide it by two, you get the next largest factor and so forth. And when you hit an odd number, you round the product up. It might just be me, but I find that fascinating.
I am not a math nut but I love this channel. Every video is crazy interesting. Keep em coming!
the largest perfect number we know is 1 million....... digits long
+joe chu according to wikipedia the largest as of this video is actually nearly 26 million, the 48th one is 34.8 million digits long (2013), the link he found was apparently in the 90s
+Kien D. N. No, the number was found in 2013 as you stated. This video was posted in 2011.
XD
+joeloud1 In 2011, the largest perfect number was 25,956,377 (found in 2008). This year, the largest perfect number is 44,677,235 digits long. Regardless, the size of these numbers is unimaginably large.
69 likes, 69! Is the biggest number you can factorial on a calculator.
I think it's interesting that every perfect number has got 6 or 28 in the final digits. In particular, the perfect numbers after 28 have got 6 or 128 at the end
So... I guess that means no odd perfect numbers?
They end in 16, 28, 36, 56, 76 and 96. This is for even perfect numbers only. Not for odd perfect numbers.
It seems you can construct a perfect number in the following way: You construct a prime number by adding 1 to a summation of powers of 2 (e.g: 3, 7, 31, 127) you proceed to multiply it by the largest power of 2 used for its creation.
For 8128:
(2 + 4 + 8 +16 + 32 + 64) + 1 = 127
127 x 64 = 8128
+Rik Cloesen it is called Euler's totient function
That is one trick I have known for a long time. Still a nice reminder tho.
I found a different pattern for generating perfect numbers but it’s difficult to explain
Thank you sir
Number theory has got to be the most fantastically mind-blowing area of math. The mere fact that one can prove theorems about numbers the definition of which I would say is basically arbitrary (i.e. perfect numbers) is absolutely beyond me.
We can exclude base numbers (x^y)
Let's do 8192 (2^13) as an example. If we add up all of its divisors, we get 8191, AKA (2^13)-1. This is true for all binary numbers, just look at the way we write them
This fact would also be semi-true for any number system, not just binary. Let's do another example: base 5. If I write 25 in base 5 I'll write it as 100. Then take the divisors and add them together (5+1=6). You can try this for yourself.
Conlusion: We can exclude every interger that can be written as x^y
By applying restrictions such as the definition of perfect numbers, we derive a finite set that is contained in the infinite set of integers. Same idea as if we applied the restriction that all integers in our set had to be greater than 0 and less than 10. We would have another finite set (1,2,3,4,5,6,7,8,9) contained in the infinite integers. Maybe I'm not understanding your question properly? He said they don't know yet whether there are finitely or infinitely many perfect numbers.
no, because the gap between the perfect numbers gets larger and larger, eventually you may reach a point where there is no larger than that. for example, 2^4 and 4^2 = 16. these are the only two integers known to exhibit that property. im sure numberphile and many people around the world (including me) would be very impressed if you found another.
3:51 rather small number for our technology
3:58 I take that back How the heck do we know that?
Hell yeah, bringing math on da street since....
well, you do the math.
116 likes from 7 years ago with no replies? Don't mind if I do...
fun fact:
8128 apears in π
somewhere between the 100th and 200th decimal digit
James: "The biggest perfect number is one million..."
Me: ok, It's not that big, I think I can handle it
James: "digits long"
Me: "oh lord..."
Lol
By the way, with that web page you went to on your phone for the largest known perfect number, I went to it on my desktop which has a larger than usual monitor and time from the instance I started holding down the "Page Down" key to the instant it reached the end of the page, and it was 41.41 seconds. That was merely holding the page down key without reading anything.
I like it when someone says "what's this number" and they look it up and say "oh its like a million digits long so"
that happens a lot on this channel...
The relation between Mersenne Primes and Perfect Numbers explained on the "31 and Mersenne Primes" video indicates that there aren't odd perfect numbers, as long as that relation stays true (for every Marsenne Prime, there's a Perfect Number that can be found through the formula {M[M+1]/2}). [M+1] is always a power of 2 that's bigger than 2 and that means the number you get from that formula will always be even.
In the formula, M stands for the Marsenne prime
+Augusto Schmitt No it doesn't. This only says every Mersenne prime produces a perfect number.
The reverse, that perfect numbers can be expressed in this way, is only true for perfect numbers that are even.
primes.utm.edu/notes/proofs/EvenPerfect.html
This doesn't say anything about the existence of odd perfect numbers.
***** Yeah, I didn't notice it when I wrote the comment, but Cooper already pointed that out
+Augusto Schmitt In a different comment? Because I can't see any other reply to yours.
***** Really? It's just above your reply. That's weird.
Here's what he said: "That does not prove that this formula creates the only perfect numbers. Odd perfect numbers would not violate this, a flaw in that rule would be computing M^2/2 + M/2 and not getting a perfect number if M is a Mersenne prime."
I just did some googling. 5 years later. It looks like they've only come up with 2 more. wow.
Well it is 45million ... digits long :) so it's only fair that only two have been discovered. Every following number seems to jump quite a bit away from the previous one.
We need more distributed computing nerds :)
Today is Oct 05th of 2018 and there is 50 perfect numbers discovered.
Well it is an np-hard problem It makes sense for the search to be slow
@@memetron3000IA 51 now!
You can calculate perfect numbers with the following formula:
2^(n-1) * ( 2^n - 1 )
where "n" is one of a very short list of prime numbers that can be used to create Mersenne prime numbers.
2^1 * ( 2^2 - 1 ) = 2 * 3 = 6
2^2 * ( 2^3 - 1 ) = 4 * 7 = 28
The first 19 perfect numbers fit this same formula with "n" values of:
2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, and 4253
oh I remember when my uncle taught me about perfect numbers, I was working so hard to find other perfect numbers
Actually, he said at the time of the video that there were 47 known perfect numbers. The perfect number he mentions that is 1,791,864 digits long is only #36 on the list and was found back in 1997. The 47th perfect number is 25,956,377 digits long and was found in 2008. The 46th was found in 2009 and was the latest known of the 47. For reference, check out: en[dot]wikipedia[dot]org/wiki/List_of_perfect_numbers
However many Mersenne Prime numbers we know of, we know of that many perfect numbers, since they're a 1:1 correspondence [to-date]. And I believe there was even a proof on another video that for every Mersenne Prime there is a corresponding perfect number.
So, we should be up to like 50 now? I think?
Did anyone else notice the video length?
I think I'm in love with perfect numbers, they're so interesting!
What is the significance of these perfect numbers? What’s their usefulness?
The reason we "only" know 47 perfect numbers is that one factor in every perfect number is a Mersenne prime, which is of the form 2^n - 1. As these primes get larger and larger, their corresponding perfect numbers also get massively large, requiring weeks or months of computer calculations to verify. While it is well-known that there are infinitely many primes, it is not known if there are infinitely many Mersenne primes (I suspect there are). But we may reach the point where we cannot find any more perfect numbers due to reaching the limits of calculating technology.
"The largest perfect number is 1 million, 791 thousand..."
Well that isn't so big.
"...digits long."
Oh. Whoa.
I thought the same, lol
I was about to comment that 🤣
It might just be me, but I see some resemblance with the powers of 2 and perfect numbers... 496 looks like 4096 (2^12), 8128 has 128 (2^7) in it, 28 is included in 8128, and, if you remove the 28 from 8128, you get 81 which is 9^2. If you remove the first perfect number, 6, from 496, you get 49 which is 7^2... What a coincidence.
Let's have 496
4+9+6 = 19
1+9 = 10
1+0 = 1
3 is my favourite number 3×1 = 3
ILLUMINATI !
the 6 is double perfect because also 1x2x3=6
Emilio Zorrilla thats kind of the point. They're factors
just because they're factors doesn't mean that those factors multiplied is that number. 12 for example has 1 2 3 4 and 6, 1x2x3x4x6=144
niekianie That's pretty interesting too, the factors of 12 multiplied together equals 12 squared. How many other numbers are like that, I wonder?
niekianie I know, that is why I said the fact that 6 is 1+2+3 and 1x2x3 at the same time is remarkable.
Emilio Zorrilla I should have said @klik, i was commenting on his reply
Way out of date: the largest perfect number now known has 49 million digits. About 30 years ago I wrote a program on an IBM AS400 to produce a list, and left it running over the weekend: it produced the first 7, but then couldn't cope with the size of the numbers :)
the first 6 or 7 perfect numbers can also be expressed by this formula, 2^(2p-1) - 2^(p-1) where p = the prime numbers 2, 3, 5, 7, 11, 13
You make math seem like fun... that is a rare talent (Y)
I love all your videos.. keep making more :D
⁹⁸⁹⁶⁰⁷²⁶⁰⁴.....
Math is fun if you understand it even the parts of it that are complicated when you understand it It's juat a game
Very informative!!!!!
Very surprised to find others who are math geeks like me on yt
Legend says he's still scrolling...
this video inspired me to create a program in c++ that checked a number to see if it was perfect or not. so far it works great! it's just very time consuming to check the large numbers, as it needs to check all the numbers less than or equal to half of it to see if they are divisors. other than that, it was fun to think about and program, and really didn't take all that long.
Nice find! So for 8128 it's kind of 127! but for addition, not multiplication, right? An easy way to write that is (n(n+1))/2, so (127*(127+1))/2 which is 8128!
4:20
Blaze it
a number more perfect than any other
4:20 is the video length. 😂
I think 6 is extra perfect, cos 1*2*3=6 too
Thomas Uijen cant say "cos" because that's another type of math 😃
cos of 6 doesnt equal 6 😂😂😂😂😂
Thomas Uijen cosine lol
another fact... all the perfect numbers that we know of either end in a 6 or a 28...
nope, there are perfect numbers that end in a 16 and in a 36 too
@@sov5257 hence they end in 6 but not 28.
@@sov5257 they end in 6
I thought that they just ended in 6 or 8!
They all end on 6 and 8
Why do people dislike stuff like this i understand dislikeing something if it's like mean or hateful. But this is so positive and seemingly helpful, even if math isn't your cup of tea.
Weird.
Hello. Welcome to the Internet.
This video is so old that there were only 47 discovered perfect numbers. Now there are 51. 4 extra in 12 years.
anyone know or even hear about postal services, FedEx initially, using these numbers to calculate optimum stops to create a route? I literally overheard this once, and sort of vaguely understood the explanation at the time, but now I can't seem to.
janglestick that... Doesn't make sense
Actually, the biggest perfect number was 25,956,377 digits long when this video was uploaded.
45.7 million
well, the biggest perfect number might be inifnitely large
This video was posted in 2011 by then a nearly 26 million digit perfect number was discovered already. In 2016 the largest one know is 44.7 million digits long.
I doubt they'll just stop finding new ones.
Probably not, as computers become better larger and larger numbers will be discovered.
And who knows with a large enough sample set a pattern could possible be discovered for finding numbers in the sequence.
+Ahmad Amin I don't even know the point but I'm facinated. I always liked basic math and doing it in my head, but I had to become 26 years old and find this channel to get into the advanced stuff. Thanks Numberphile
Nope, new perfect number 46M digits long found recently.
Yes these are known as Mersenne prime exponents. You have 2, 3, 5 and 7, the next few are 13, 17, 19, 31 and so on up to the 48th which is 57885161. So that would make 57885161 leading 1's and 57885160 0's.
The simpler equation rather than converting a binary number that large is:
(2^(p-1))*((2^p)-1)
So for example, for 8128 the Mersenne prime exponent is 7, so:
(2^(7-1))*((2^7)-1) = 8128
This is slightly simplified due to youtube constraints, but that's the general idea.
As of January 2016, 49 Mersenne primes are known, and therefore 49 even perfect numbers (the largest of which is 2^74207280 × (2^74207281 − 1) with 44,677,235 digits).
There's a pattern. The length of the subset of powers of two doubles. The next element in the set is the next power of 2 - 1. The succeeding terms are simply double the previous. Whether that's the only pattern that yields perfect numbers is unknown.
Hahaha! xD That was my reaction exactly! I don't know if he did it on purpose, but it looked like he did. xD
Ah, thank you for checking this and explaining it. Now it just makes me wonder if there are other perfect relations in other base counting systems, and if they are more numerous than in base 10.
This does not depend on the number system in use. We only use addition of divisors here.
So that means that the current largest perfect is between 10^1,000,000 and 10^10,000,000
Now, in 2016, there are 49 perfect numbers (source: en.wikipedia.org/wiki/List_of_perfect_numbers)!
In a book I am reading "Excursions in Number Theory" the author mentions that all perfect numbers end in 6 or 8. However, he doesn't provide a proof, yet, maybe it's in a later chapter.
What about between 10,000 & 100,000?
8 is perfect it's symmetrical horizontally and vertically *does anyone get that soul eater reference*
I did, but then you ruined it by asking if I did...
:c
+otaku nub Aww, don't be sad! I'm sorry for being mean. [headpat]
c: thankyou
0 is too
A perfect number has to be even, as a single number, the highest factor excluding itself, must be half of the number. This must be so, because it cannot be larger than half, because that is the number paired with two when multiplying. If it were not even, you would not have that number, and then, the smallest possible factor other than one would be 3, and one third of the number would not be enough to cover the massive gap in value between the perfect number and the sum of its factors.
945 (not even) is the smallest odd number that of which we would call "abundant". Not only is the sum of it's factors able to make up the gap without two, but it goes over.
Also, what about the relationship between perfect numbers and powers of two... 6=(2^3)-(2^1); 28=(2^5)-(2^2); 496=(2^9)-(2^4); even 8128=(2^13)-(2^6)... There are so many patterns there that I'm not going to even mention them... If it's not proven that all perfects are even, it certainly seems so based on this trend.
So 5!= 5×4×3×2×1
And 0! = 0×0=1?
5 is a prime number you bampot
Oh, another old comment thread that got a new comment. It's been a while since my last comment (141w).
A lot of people brought up the "proof" that 0! = 1 by explaining that if you divide N! by N, you get (N-1)!. Such as:
5! (5*4*3*2*1) / 5 = 4! (4*3*2*1)
However, using this as proof breaks down once you go below 0. Sure, you can use it and do: 1! (1) / 1 = 0!, but that shouldn't matter, it's not proof. If you go into negatives, then this method breaks down too. 0! / 0 = - 1! = Undefined.
n! can only be expressed as n*(n-1)!, when not dealing with 0. Think about it, sure, 1! = 1 * 0! which equals 1! = 0!...but what about negative numbers?
0! = 0*(0-1)! or 0! = 0 * -1!
If 0! is supposed to equal 1, then how can 0*-1 = 1? Anything multiplied by 0...is 0. You might say that it doesn't work for negative numbers, I would say that is true and it also doesn't work with 0.
Proof that doesn't work with every number, in this case, isn't proof. If the definition of a factorial itself doesn't include every number, it starts working properly. n! where n > 0.
Another line of reasoning is factorial being the number of ways you can arrange something. You can arrange 3 objects 3! ways (6). According to them, you can arrange 0 objects 1 way. It makes sense, sort of. You can use this as a tool for this kind of thing, but it isn't the definition.
Lastly, factorials are just multiplying a number by the number below it, and the number below that...before you reach 0.
5! = 5*4*3*2*1, 100! = 100*99*98*97...*1
This can be worked out like this:
n! = n * (n-1) * (n-2) * (n-3) ... * 1
Basically until n - x = 1.
If you put 0 into this, it cannot possibly work. x increases in value, so you cannot use 0 - (-1) to reach 1.
It has to stop at 1.
3! = (3*2*1), 2! = (2*1), 1! = (1), 0! = ( )
0! = Undefined.
EXCEPT mathematicians have agreed on the definition themselves, even if there is no concrete proof for such a definition. The only proof is just a coincidence. It's a method that works for ALMOST every number.
There are no decimal factorials (1.5!), no negative factorials (-1!) and no 0!.
Anyway, sorry the for the rant-ish comment, I wanted to cover a lot of the replies and I wanted to post an update on my opinion after all this time. After all, 0! equalling 1 doesn't affect much, but it shouldn't be something we all agree on because some incomplete proof says so.
I did not expect this.
.
CrazySkullGamer what are you saying??
why am I watching this
Why is 1 not a perfect number? It is divisible by 1 and nothing else, so, 1+nothing=1, right?
TheGamingBros 1x1x1x1=1 infinitely add it
well for the ones listed so far, the first digit is either 6 or 8. that could help^^
Well now, almost 6 years after this video, the largest known perfect number is 44,677,235 digits long. As in we now know 49 perfect numbers.
I was wondering that but James contradicts that near the end of the video
3:45 As the 47th perfect number (the largest one known) is 1,791,000 digits long, it cannot follow that pattern forever
If it did follow that pattern, it would only be 47 digits long (still fairly large though)
6 in binary is = 110
28 in binary is = 11100
496 in binary is = 111110000
8128 in binary is = 1111111000000
So every perfect number is written in binary with (n) ones and (n-1) zeros?
Yeah. There is a pattern...
No. 5. 33.550.336 == 0001 1111 1111 1111 0000 0000 0000
No. 6. 8.589.869.056 == 0001 1111 1111 1111 1111 0000 0000 0000 0000
No. 7. 137.438.691.328 == 0001 1111 1111 1111 1111 1100 0000 0000 0000 0000
For perfect numbers, it looks like numberphile went to the hood!!!
We now have 48! There was another perfect number discovered in 2013 and it is 1.8m digits long!
Numberphile and brown sheets of paper. The perfect duo.
Edit: Perfect numbers are of the form 2^(n-1)x (2^n-1) [Mersenne prime], but there is no apparent infinite series based on variable n.
so, the question is - is there any pattern between those? can we say that there is any function that actually make those perfect numbers?
by the way, they don't divide by each other (28 doesn't by 6, 496 doesn't by 28 or 6 etc.)
Perfect is just a name given to these numbers. We know nothing is perfect, but it's just a name for this set of numbers.
After googling ,the formula for finding perfect numbers is found to be N=2^(p-1)(2^p-1), where p & 2^p-1 are both primes. Amazing, right?
The website he looked it up on is out of date. :) The perfect number with 1,791,864 digits was discovered in 1997.
The most recent one at the time of this video was 25,756,377 digits long. The most recent one currently is 34,850,340 digits long.
That's fun to even think how does computer calculate numbers that take a whole Terabyte HDD to write down when you need space for each such number tested to write down as checked and not perfect. And then, as computer can glitch, human must check it by manually finding all divisors and adding them up XD
'The largest perfect number is 1 million...'
Me: 'Okay...'
'Seven hundred...'
Me:'Okay...'
'And ninety one thousand...'
Me:'Riigght, not too big then?'
'Digits long.'
Me. ' O.o.... '
That's what I meant: The way of answering this question is looking out for a pattern of perfect numbers so that they can be determined without "manually trying 'em out". And yes, I know that ain't easy.
For apparently "perfect" numbers, they sure aren't very elegant! Highly-composite numbers are the real heroes here, especially because they're actually useful.
So I was thinking that there can't be odd ones because you need the 1/2 for any possibility of the summation series reaching the original number (otherwise you never add up to more than 1/2 the original number). I am assuming that is what your equation means but I'm not sure.
If you make the rules ,I'd say you can answer the questions you say we don't know. This only works if you use simple numbers too, designed by us to count whole amounts.
at 2:40 there are 51 :P (in 2019, kind of fun to watch these videos after 8 years...) Edit: and of course, you wanted to say "25 million... digits long" (for the 47th). The one going over a million digits was the 36th, discovered in '96 (1.7 million digits). Currently, we reached close to 50 million digits with the 51st discovered
you dont do that you use recursion, to program the data into a word processor viola. you just need to think of the base states then
I have'nt read all the entries, so i may repeat someone else. This is all about the decimal system. What about other systems, like 8 of hexadecimal? Is there also a definition for "perfect" in those systems?
Oh golly back in 2011 things were rough for Numberphile.. they used to make videos under a decript back alley.. with a small brown paper...
Look at how that iPhone 4s has aged! That tiny screen we all were so happy with.
James: "The largest known perfect number is, apparently, 1,791,000..."
Me: "Oh, really? I thought it would be a bit-"
James: "... digits long..."
Me: "WTF!?"
The length of this video is a pretty perfect number
It's not a fight, it's an agreement and furthering eachothers work. We call this collaboration, Borat. Welcome to America!
at some point i was bored and found that after some calculations 2520 is a very fun number, as it is devide-able by every number between(and including) 1 to 10
47 perfect numbers were found? Thats probably the answer to all our questions... :D
Perfect number :- sum of factors of number is double the number.
Form 2^(P-1) × 2^P - 1 ; P is a prime no.
Ah, yes. Thanks! My bad.
In my defense, it was late at night and I was tired...
Aww, who am I kidding! That's no defense!
Given that there are now over 2000 comments one this video, I have no idea what this means. So naturally, I liked the comment.
on a different but not unrelated note, i'm not sure why he and wikipedia state that it is unknown whether there is any odd perfect numbers. correct me if im wrong, but the only way you can get a perfect number is by first finding a mersenne prime (always odd) and multiplying it by itself (+1) (therefore always even). if you multiply any number by an even number, you get an even number. (which then of course you divide by 2 to get an even number (your perfect number))
Interesting thing (or maybe not): in the largest perfect number, no digit repeats more than 7 times in a row.
All the known perfect numbers are of the form (2^(p-1))*((2^p)-1), where (2^p)-1 is prime. It's called a Mersenne prime. Every Mersenne prime generates a perfect number in this way, but most numbers of that form are not prime, and there is at present no way to know which ones are prime except to actually check them.
this is derived by eulers identity that if 2^k-1 is a prime number than the perfect number is 2^(k-1)(2^k-1)