# 8128 and Perfect Numbers – Numberphile

BRADY HARAN: So we’re going to
talk about the number 8,128. And it’s a famous type of number
called perfect number. Now, this was the largest
perfect number known in antiquity, or for
very long time. But what is a perfect number? The smallest perfect number
is the number six. So let’s look at
the number six. Now what does it mean? What divides into six? Well one divides into six. That’s always true. Two divides into six. And three divides into six. Now, six does divide as well,
but for what we’re going to do, we’re going to ignore
the number itself. So ignore six. But if we add up the numbers
that divide six– one, two, and three– add them
together, and we get the number six. We get the number itself. So this is a perfect number
because the divisors add up to the number itself. It’s not too small, and
it’s not too big. It’s just right. It’s perfect. And in antiquity, they
were given this rare, perfect property. So the next one, after
6, is the number 28. Let’s write that down. 28. So what divides 28? Add these together, and
you get the number 28. It’s the second perfect
number. The third perfect number after
that, bit of a gap after that, the third perfect
number is 496. Let’s look at the divisors
of this. And we’ve got 1. Quite a few of these. 2, 62, 124, 248. We’re running off the edge. We add all this together in
the same way as before. We get 496 again. It’s perfect. Now, there’s only one perfect
number between 1 and 10. That’s the number 6. There’s only one perfect number
between 10 and 100. That’s 28. There’s only one perfect number
between 100 and 1,000– 496. And there is only one perfect
number between 1,000 and 10,000, and that is 8,128. Let’s try it out then, OK? 1, 2, 4, 8, 2,032, and 4,064. Add those together in the same
way, and you will get 8,128. Today, even with our computers
that can compute massive numbers, we have only found
47 perfect numbers. That is all. Four were known to the
ancient Greeks. Still today, 21st century,
47 we’ve got. There may be more. There may not be. We don’t know if there’s
an infinite many of these numbers. And that’s an open question. You’d think that that’s
something we would know by now. Are there infinitely many of
these or aren’t there? We don’t know the answer
to that yet. MALE SPEAKER 2: What’s the
biggest perfect number? BRADY HARAN: So far? I wouldn’t know that. Let’s find out the largest
perfect number. Well, Wolfram Alpha says, do
you mean largest integer? There’s no largest integer. So that’s helpful. Interestingly, we don’t know
if there are odd perfect numbers either. All the perfect numbers we
have found are even. That’s another question
we don’t know. The largest known perfect
number is, apparently, 1,791,000 digits long. Don’t ask me to say it. We could be here
’til next week. But that is the full thing. Nearly 2 million digits long. I could be scrolling through
this for quite some time.

### 100 thoughts on “8128 and Perfect Numbers – Numberphile”

• December 8, 2016 at 8:24 pm

why is 8128 special than the rest of 47 numbers ?

• December 11, 2016 at 12:53 am

You wouldn't be able to say million digits in one week đ

• December 22, 2016 at 1:12 am

Did anyone else notice the video length?
I think I'm in love with perfect numbers, they're so interesting!

• December 30, 2016 at 4:49 pm

.there is an infinite amount of them because that is the nature of infinite numbers

• January 1, 2017 at 5:16 pm

@4:20

• January 6, 2017 at 11:11 pm

the smallest perfect number is 1. 1 divides into 1, 1 = 1

• February 15, 2017 at 11:14 pm

why 1 is not a P.N?

• February 22, 2017 at 6:33 pm

Really sorry if I am being dense, or I have missed something already commented on: all the Perfect numbers (seeme to be) triangular, based on Mersenne primes. Is this true; is it provable, and if so, surely our knowledge of Mersenne primes should give us many more 'Perfects'…?

• February 23, 2017 at 3:18 am

Gonna say, if there's an infinite number of numbers then there's an infinite number of perfect numbers along with the following
Powers of 2
Decimal
Square root

• February 24, 2017 at 5:25 am

blaze it 420 length

• February 26, 2017 at 6:15 pm

1 is a perfect number right?

• March 1, 2017 at 8:56 pm

I like these videos but the application of this information in the real world is probably nonexistent. So basically most of the work that mathematicians do is irrelevant. Unless it does have some application which would be cool.

• March 3, 2017 at 9:43 pm

2 more have been discovered since this video was made making it 49 perfect numbers in total so far

• March 8, 2017 at 8:30 pm

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.

• March 19, 2017 at 8:15 am

is 120 a perfect number?
120 = 1,2,4,8,15,30,60
and 1+2+4+8+15+30+60 = 120

• March 25, 2017 at 7:53 am

The length of the video is a perfect number too….. 420

• March 29, 2017 at 7:33 pm

420 blaze it, that's the most perfect number

• March 29, 2017 at 7:49 pm

there is a perfect way to find perfect number. They r always even numbers. There r infinite many numbers. the formula is – EVERY NUMBER WRITTEN IN AS A PRODUCT OF 2 TO POWER (n-1) and 2to power N -1 is a perfect number where n is a prime number greater than 1. In the formula, if we take n=2, we get 6. If we take n=3, we get 28 and so on…Thank you

• March 30, 2017 at 6:00 pm

I wouldn't call a number perfect, unless it had symmetry.

• March 31, 2017 at 2:22 pm

So isn't one a perfect number?

• March 31, 2017 at 11:12 pm

Oh, this is silly. Get off it.

• April 2, 2017 at 5:17 pm

but wher is the 4 ??

• April 2, 2017 at 5:19 pm

Well, I guess the 1 doesnt count per definition?

• April 9, 2017 at 9:21 am

It strikes me as very easy to prove that odd numbers can't be perfect. Each of their factors are a maximum of 1/3rd their size, and the most "efficient" numbers, those with only single instances of prime factors, all seem to fall far short when summing their factors.

• April 15, 2017 at 4:58 am

"1 million" ok like 7 digits long ok "Digits" my jaw is on the antipode of me

• April 25, 2017 at 12:15 am

There are infinite perfect as there are infinite numbers.

• April 30, 2017 at 7:39 pm

1+2+3=6, 6 divides into 3
1+2+3….+7= 28, 28 divides into 7
1+2+3…+31=496, 496 divides into 31
1+2+3…..+127= 8128, 8128 divides into 127
I think I see a perfect number proof :O

• May 4, 2017 at 2:08 am

Wouldn't 1 be perfect cuz 1=1

• May 16, 2017 at 12:46 pm

I will try to find the next one with paper and pen!

• May 21, 2017 at 12:39 am

if you found a new one write in comments :))

• May 21, 2017 at 2:39 am

To my own knowledge, there couldn't be an odd perfect number, could there? To get anywhere close to the number, there would have to be at least half of the number itself.

• June 3, 2017 at 10:17 am

Anything so close to but not quite a power of 2 triggers me. I have OCD and I'm a programmer.

Just kidding

• June 22, 2017 at 4:16 am

Does the study of Partitions help with finding Perfect numbers?

• June 22, 2017 at 1:57 pm

Just me or do they all end with 6 or 8?

• June 22, 2017 at 9:11 pm

Either im too high for reality or this video is actually 420 stuff long.
Counted and lost count, going to sleep.

• July 25, 2017 at 1:23 pm

The lenght of this video is another example of a perfect number. Well played Numberphile.

• August 1, 2017 at 5:32 pm

James was wrong; at the time of this video, the largest known perfect number was 25,956,377 digits long. The number that he found, which was 1,791,864 digits long, was discovered in 1997, and is currently believed to be the 36th perfect number (which is to say, it is believed that there are exactly 35 perfect numbers smaller than it).

• August 10, 2017 at 9:02 pm

Doesn't the number have to be a triangular number also? This is what I was taught – it may not be true.

• August 31, 2017 at 7:01 pm

Odds numbers are not perfect.poor odd numbers

• September 26, 2017 at 12:17 pm

I created what I thought would be a new formula to finding a perfect number (obviously it had to involve a mersenne prime), and by induction, I was able to prove that;

P = 2^(2n – 1) – 2^(n – 1) iff P = 2^(n – 1)(2^n – 1)

Is this a new way of finding a perfect number or has it already been discovered before? More about my formula is on the Mathematics Stack Exchange.

• November 4, 2017 at 4:40 am

There are no "perfect" numbers. All numbers are perfect in their own way. Stop trying to impose your antiquated rules on what a number should be.

• November 8, 2017 at 4:48 am

Wouldnât 1 be a perfect number?

• November 11, 2017 at 12:57 am

Of course the ancients would stop at 10,000 â âmyriadâ to the Greeks.

âFrom one thing, know ten thousand things.â is a famous quote from Miyamoto Musashiâs âBook of Five Ringsâ, a martial arts classic that goes well beyond the martial arts.

• November 18, 2017 at 5:10 pm

One?
Maybe?

• November 24, 2017 at 6:44 pm

I want numberphile to review my video… I have a discovery involving exponents that will blow the minds of the best mathematicians.. please watch my video

• November 28, 2017 at 10:19 pm

Happy 6th birthday to this video!

• December 12, 2017 at 1:10 pm

isnt 1 a perfect number?

• December 12, 2017 at 2:00 pm

who id from 2017? largest perfect number is 44,677,235Â Digits long)

• December 15, 2017 at 7:54 am

is anyone else unable to see or reply to the comment chain of the guy showing the powers of two perfect numbers

• December 23, 2017 at 6:15 am

0 is left out đ

• December 25, 2017 at 10:07 pm

4:20 perfect number

• December 25, 2017 at 10:10 pm

So only works in base 10

• December 27, 2017 at 12:17 am

why isn't 1 considered a perfect number?

• December 29, 2017 at 5:58 pm

I have to disagree here. All factors come in pairs so if you're discluding 6 you should also disclude 1. A perfect number should actually be defined as a number whose factors all add up to 2 times the original number therefore there's no disclussion of any factor. Choosing just one factor to disclude means that you could say that about any number by just discluding one of the factors. It just doesn't sit right with me. (And yes even square roots come in pairs, it's just two of the same number)

• February 14, 2018 at 7:29 am

All perfect numbers are divisible by a square including odd numbers.

• February 16, 2018 at 5:43 am

Switching between addition and multiplication, really has no meaning, and only create artefacts. I'm afraid people doing that don't understand Mathematics at all

• February 23, 2018 at 2:17 pm

haha the captions thinks James is Brady and the cameraman is just called âMale Speaker 2â

• February 27, 2018 at 8:32 am

I feel like I want to object. It seems rather imperfect to include 1 but not the n itself. Either have both or neither. o.o

• March 1, 2018 at 4:12 am

Proof that all perfect numbers are even, going by what you said in the Mersenne Primes video:
(Perfect_n)=((Mersenne_n)*((Mersenne_n)+1))/2=(2^a-1)(2^a)/2=(2^2a-2^a)/2=2^(2a-1)-2^(a-1), which is even for all integers a>1, a is such that 2^a-1 gives the nth mersenne prime.

• March 14, 2018 at 10:16 am

• March 15, 2018 at 11:01 am

The perfect numbers seem to all end in 6 & 8

• March 20, 2018 at 5:48 pm

Is there any practical applications for perfect numbers?
I was a sub teacher. a student asked me is there a practical application for rational and irrational numbers? I said yes in mechanical engineering the design of gear trains.

• March 20, 2018 at 5:57 pm

how was the largest perfect number found? by trial and error or is there a formula?

• March 21, 2018 at 10:02 am

Shouldn't the smallest perfect no. Be 1 1=1 and 1=1 both cases true

• March 25, 2018 at 10:30 pm

Would love to show the ancient Greeks a number almost two million digits long đ

• March 30, 2018 at 9:10 pm

6 is also a factor of 6… When you add it on it becomes 12…

• April 2, 2018 at 8:22 pm

• April 7, 2018 at 6:09 am

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.

• April 20, 2018 at 2:27 pm

You can see that it's an old video because the iPhone won't bend

• June 6, 2018 at 7:34 pm

• June 17, 2018 at 3:58 pm

There are 50 perfect numbers now and the largest one consists of 46,498,850 digits

• June 25, 2018 at 8:05 am

we never have more than 3 dimensions in the number system why 2^3 is 8

• June 25, 2018 at 8:49 am

3Ã4 three d people in a four d world

• September 11, 2018 at 10:48 pm

Comment.

• September 21, 2018 at 2:26 am

Video is 4:20 long

• October 20, 2018 at 8:23 am

He is using an iPhone 4s… I am also watching this video on iPhone 4s. Wow

• October 31, 2018 at 1:17 am

Aftet the first one if you add all the digits up
2+8=10
8+1+2+8=19. 1+9=10
it keeps going with each one =10 everytime

• November 9, 2018 at 10:08 pm

There are now 50 known perfect numbers. There is a one-to-one correspondence between Mersenne primes and even perfect numbers, so whenever the Great Internet Mersenne Prime Search search finds another Mersenne prime, it gets a new even perfect number for free.

• November 16, 2018 at 10:01 pm

The legend says he's still scrollin' today..

• December 17, 2018 at 10:33 pm

puurfact nombah…

• December 31, 2018 at 8:44 pm

But does the sequence of only one perfect number between powers of ten hold true past 10^4?

• January 23, 2019 at 4:03 pm

How many perfect numbers do we know about now in 2019Âŋ

• January 26, 2019 at 1:42 am

THERE ARE NO PERFECT NUMBERS!
Every number has an a factor of itself

• January 29, 2019 at 9:09 am

Numba

• February 22, 2019 at 5:52 pm

In primary school our teacher said to us "find the 5th perfect number"

I still do nightmare about it đ

• March 8, 2019 at 9:55 am

The marker being kinda out of ink makes me nervous

• March 31, 2019 at 6:03 pm

"we don't know if there's infinitely many of these numbers"
i've heard that so many times in numberphile videos and i don't get it??? there are infinitely many numbers, so within that, there must be infinitely many of certain types of numbers… am i missing something?

• April 3, 2019 at 9:08 am

Thanks for its I understand perfect number

• April 14, 2019 at 8:36 pm

This video was published on November 28, 2011, and at the time, the largest perfect number was (2^43211609 – 1) x (2^43211608). This is the 47th perfect number, found in 2008 and is 25956378 digits long. But what James said is the 36th perfect number, which is (2^2976221 – 1) x (2^976220). That was found in 1997 and is 1791864 digits long.

• June 18, 2019 at 5:09 am

throwback to the iphone 1

• July 11, 2019 at 12:04 pm

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

• July 17, 2019 at 8:50 am

Non-prime factors are boring :-/

• July 22, 2019 at 3:03 am

All perfect numbers are triangular numbers

• August 20, 2019 at 6:03 am

The biggest pefrect number as of 2018 is 46 million digits long

• August 22, 2019 at 11:22 am

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 đ

• September 7, 2019 at 1:25 pm

at 2:40 there are 51 đ (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