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.

why is 8128 special than the rest of 47 numbers ?

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

Did anyone else notice the video length?

I think I'm in love with perfect numbers, they're so interesting!

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

@4:20

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

why 1 is not a P.N?

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'…?

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

blaze it 420 length

1 is a perfect number right?

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.

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

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.

is 120 a perfect number?

120 = 1,2,4,8,15,30,60

and 1+2+4+8+15+30+60 = 120

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

420 blaze it, that's the most perfect number

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

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

So isn't one a perfect number?

Oh, this is silly. Get off it.

but wher is the 4 ??

Well, I guess the 1 doesnt count per definition?

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.

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

There are infinite perfect as there are infinite numbers.

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

Wouldn't 1 be perfect cuz 1=1

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

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

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.

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

Just kidding

Does the study of Partitions help with finding Perfect numbers?

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

Either im too high for reality or this video is actually 420 stuff long.

Counted and lost count, going to sleep.

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

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).

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

Odds numbers are not perfect.poor odd numbers

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.

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.

Wouldnât 1 be a perfect number?

Of course the ancients would stop at 10,000 â âmyriadâ to the Greeks.

âFrom one thing, know ten thousand things.â is a famous quote from Miyamoto Musashiâs âBook of Five Ringsâ, a martial arts classic that goes well beyond the martial arts.

One?

Maybe?

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

Happy 6th birthday to this video!

isnt 1 a perfect number?

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

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

0 is left out đ

4:20 perfect number

So only works in base 10

why isn't 1 considered a perfect number?

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)

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

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

haha the captions thinks James is Brady and the cameraman is just called âMale Speaker 2â

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

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.

Take âBoincâ – upload for free – on the PRIME GRID Projekt

The perfect numbers seem to all end in 6 & 8

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.

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

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

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

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

What about zero?

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.

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

what about NEGATIVES

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

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

3Ã4 three d people in a four d world

Comment.

Video about perfect numbers

Video is 4:20 long

Tq

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

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

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.

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

puurfact nombah…

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

1282

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

THERE ARE NO PERFECT NUMBERS!

Every number has an a factor of itself

Numba

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

I still do nightmare about it đ

The marker being kinda out of ink makes me nervous

"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

mustbe infinitely many of certain types of numbers… am i missing something?Thanks for its I understand perfect number

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.

throwback to the iphone 1

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

Non-prime factors are boring :-/

All perfect numbers are triangular numbers

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

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

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

We should give these found perfect numbers a name like "perfect 1, perfect 47" for ease of use since they are too long to say and quite frankly I do not know the suffix for such quantities.

homestuck