Perfect Number Proof - Numberphile 

Is putting the lid on a pen the maths equivalent of dropping a mic, then?

I differentiate between groups of operations not with inflection, but with pauses: "Two to the... n minus one," versus "two to the n... minus one."

Matt is great. I love his sense of humor. He's one of very few people who can take the subject of this video and make in entertaining to non-math nerds.

Matt becomes the child he talks about at 3:43, at 13:37

Classic Mathematician: "Let us assume that we know the total"

13:39 Literally the smuggest face ever :')

It's 2am. I've become addicted to watching Numberphile before bed. I'm watching towards the beginning where we are looking at the pattern of the 2, 4, 16, 64... And I think to myself, those are powers of 2. Then I see they are the prime -1. I figure Matt will say "this is obviously just 2 to the power of the prime minus one." When he says he tortures kids with it and it's not obvious at all I feel so happy that I finally understood a non obvious Numberphile concept. I finally feel like I belong. Loved this video!

Matt is the author of Things to Make and Do in the Fourth Dimension. You can support him by checking out his book... On Amazon US: bit.ly/Matt_4D_US Amazon UK: bit.ly/Matt_4D_UK Signed: bit.ly/Matt_Signed

It blows my mind how similar of a feeling this video gives me to watching my calc 2 professor do proofs for certain series tests...

Oh that was beautiful; math truly is the music of logic!

"I use this to torment young people" :)

You should do more of these proof videos, this was really great!

Matt Parker, I would have loved to have you as a math professor in school. I've always loved math, and even majored in it as college. It was teachers like you who made it even more interesting.

Matt looks so happy at the end of this video :D

Haha.. "that's why Australians are so good at math" 4:54

For some reason, I find Brady's incredulity at the beginning to be hilarious."You've already shown a link!"

I just pronounce superscripts more quickly when they're together, like parentheses

I love this channel! Matt has told me everything I needed to know about perfect numbers and mersenne primes in this video and his one that came right before it that I can teach it to my classmates that know nothing about it.

I think I would've gotten stuck at the geometric series step, but everything else was explained well and clicked for me. Cool!

I would have loved to see a proof of the other way around, that is every even perfect number has a Mersenne prime factor.

Yay! For once in my life I did the whole thing myself before watching the video. The only difference with my method was to prove the sum of that particular geometric series by induction, because I already knew what the answer was by inspection, so it seemed like the best proof to use, especially given that I didn't even notice it was a geometric series...

Fabulous proof. Thank you for a great video.

Last two vids were really good. Keep it up Brady.

Happy 2016 Matt. I enjoy your videos.

Holy frick I am blown away, this is one of the coolest things ever

I saw you on tv! Outrageous acts of science! Haha that's awesome.

Love it! I may need to watch it again! Any plans for a Numberphile book?

I haven't done math in ages, but I'm proud to say not only did I follow along with the video, but I was a step or two ahead.

An interesting property of even perfect numbers that follows this theorem (although the proof is not as exiting) is that all even perfect numbers end with the digits 6 or 28. Another interesting fact as that the proof in this video was proven in one way by Euclides and by Euler in the other, two of the greatest mathematicians of all time. Euler also did some work on odd perfect numbers.

If you've ever worked with binary you know that the sum of all the powers of 2 up to n - 1 equals 2^n - 1

I like this proof! Helped me to understand what was shown on the previous video.

"torment young people" LOL keep that up!

I love these videos!

Rising inflection,,, good work

Lovely video, thanks. This link was known to the ancient Greeks... but the converse (that all perfect numbers are of this form) had to wait until Euler. I wish you could dedicate one more video to this other side of the proof.

A new year, a new Matt Parker video. What a great start to 2015! (Although I'm sure Matt would argue that a year is a meaningless or at least arbitrary measure of time)

I saw the pattern, I've never felt so accomplished

"I'm so pleased I'm going to put the lids back onto both of the pens" Hahahaha! You're good on camera! Well proof'd :)

Matt has got to be the best teacher ever.

So what about the proof that all (even) perfect numbers are of this form?

Beautiful ❤️. Congratulations!!!

love you, matt and brady

Is it just me, or does anyone else get a real self-satisfied kick out of people who insist it's not possible to solve infinite sums in the manner described starting at 11:00?

Oh my goodness Brady is making a mausoleum channel.

thank you guys!!!!!!!

Why math > science: You dont have idiots claiming satanism in the comments. (and just to be clear im not saying everyone is like this)

13:34 turns to camera, looking very smug, "but now we've managed to prove it"...

You proved each Mersenne prime makes a perfect number of that form. You should prove the converse too: every even perfect number has that specific form.

Superb video.

Can't wait for objectivity! The onscreen links didn't work though, but the description wasn't far away :)

The power series summation came very naturally to me, because I saw it as all ones written in base b. Most easily in base 10, 10^5 + 10^4 + 10^3 + 10^2 + 10^1 + 10^0 = 111111 (in base ten which is very natural for most of us). But adding powers of 2 aren't any different now. 2^5 + 2^4 + 2^3 + 2^2 + 2^1 + 2^0 = 111111 (in base 2). The challenge in how to get to all ones with a single equation is done by going one power higher, subtracting one, and dividing by the digit repeated from the result. 10^6-1 = 999999, so divide by (10-1) to get all ones. 2^6-1 = 111111 (base 2), already ones but you can divide by (2-1) for giggles 8^6-1 = 777777 (base 8), so divide by (8-1) to get all ones. 5^6-1 = 444444 (in base 5), so divide by (5-1) to get all ones and when you have all ones, you have a sum of a geometric series. Sum b^0 to b^n = (b^(n+1) - 1) / (b - 1) This may seem contrived or odd to many people, but until I saw it this way, it never really clicked. Perhaps it is a way to show it to somebody who doesn't understand it through other ways.

3:35 "professional jerk". I'd love to see that as your profession on official documents.

Totally above my intelligence! Looking forward to next video

1st few souls to see this one!! XD don't know if it indicates how responsive my cellphones notifications are.. or how interesting these videos are that it makes me watch them even when i have a test the following day XD

And I'm screaming 256 without thinking it through, I guess I subconsciously realized it was powers of two.

..."negative one plus two to the n"...ambiguity gone

To avoid confusion it might help to be a bit more rigorous - and a bit more formal - with the syntax, differentiating the product of 2^n minus one from two to the power of the difference of n-1. It's a litte harder to follow, but if you understand it, it makes it clearer which is which.

Beautiful.

I've seen Matt Parker in countless numbers of these videos and I just realized he reminds me of The Doctor.

I wish to be at one of his classes🤓

I knew that 6 was a perfect number from my childhood, but on a lonely day with nothing to do (and before the internet) I worked out that 28 was the next one and that 496 the third one when I noticed the pattern in the factors and stumbled onto Mersenne primes by accident as I tried to work out more perfect numbers. I was so excited! Alas, that I was not the first (by millennia) - but it was still fun to discover on my own! Awesome video and explanation of why it works out this way. Thanks!

3:36 oh wow damn, im not sure if maybe i once already watched this video and forgot or already watched a video about it and forgot but i actually managed to figure out the pattern first try, kinda happy about that uwu

Isn't the pattern more clear in binary? Aren't we obscuring the pattern by thinking in decimal?

What's à perfekt Numbers? Lol Swedish spelling correction when typing English :) What's a perfect number - the perfect question to answer at the start of this video!

That was beautiful

I have been a fan of both Perfect Numbers and Mersenne Primes since high school (~50+ yrs ago!!), but I have never seen this proof! In the immortal words of Mr. Spock..."Fascinating!"

More of this on numberphile would be appreciated :) this is maths

The perfect numbers are the triangular numbers of the Mersenne primes, or the factors that you multiply by are half the prime plus 1

Watching people do math is like watching people dance - I can't do either, but it's fun to watch someone who does it well.

The largest known perfect number, which is the 51st perfect number known, is (2^82589932)(2^82589933 - 1)

I don't change my tone when differentiating between 2^(n-1) and 2^n-1. I use pauses. There's 2 to the…n minus one vs 2 to the n…minus 1.

Excellent presentation of the topics. Many many thanks. DrRahul Rohtak India

I'm in high school and I got the pattern before you said it. I do feel smug :)

Fun proof. Similar to a lot of the proofs I did when studying polygonal numbers.

Yeah! I found the Pattern for the Factors :D

Yey new video

Knocks me Mersennesless! Two-per duper! Foundational number theory I would think. And by the way, who do you think WILL win the geometric series this year? The Common Ratios are favored. Thanks! Like this a lot!

Hi Numberphile, thanks for this lovely video. At time 10:10 I look at the workings and can't understand why the first line of the calculation (on top) is multiplied by (2^n) - 1, this is not included on the second line up from the bottom, did I misunderstand somewhere?

I found the 2,4,16,64 incredibly quickly. I’m not a genius, I just had already read the top comment

In a previous video about the mandelbrot set and the numbers 63 and -7/4, Dr. Krieger stated that every Mersenne number (other than 63) would have a new prime divisor. Is there any way you could show a video of a proof of that? She also said that 63 being the 6th element in the sequence was the cause of it not having a new prime divisor. Is that because it is a perfect number? In that case, would the 28th element not have a new prime divisor as well? I've been struggle to find anything online proving her statement and I haven't been able to prove it myself either so if a video could be made (or at least if I could be given a link to an article) that would be fantastic.

About loking smug. Matt's look at the end.. :)

That smug face at the end! :D

yay more channels

At 9:16, I start seeing two sequences multiplied by the Mersenne prime -- instead of just one.

I demand a Parker prime!

@9:09 a wrong factor (2^n-1) appears in the first line

that hit me right in the maths

What I would give to have Matt Parker as my maths teacher...

Just perfect Math :)

I am in severe awe of this man's mathematical prowess.

I went through another day not having to use these calculations, again.

I immediately saw that pattern as 2^(n-1) because binary, 2^n (because of programming, binary is something I use on the daily), and because it related to the equation (2^n)-1, also related to binary. It's funny when you think about it, math and programming are so similar yet so different, or at least in my mind they are.

There's a mistake at 10:00. It should be: (1+2+...+2^(n-1)) + (2^n -1) + ......... You wrote: (1+2+...+2^(n-1))*(2^n -1) + (2^n -1) + .........

Love it 😂👌

Another way to see that the geometric sum of 2:s at the end is equal 2^n-1 is to see the sum as a strip of n-1 1:s in binary which is the same 2^n-1

The reason I can watch this video on faith is that I enjoy his enthusiasm above all. However, I would like to learn number theory from him or his recreational math theory ^_^. I'm reading Love & Math and I've never felt so stupid. But it makes me want to learn more about number theory and topology.

Matt, When I go to university I want to be in your class! What university do you teach at? I got your signed book for Christmas with shapes of constant width 2d and 3d, utilities mug, and the heart keyring! (I can't remember what it was called) they were the best presents ever!

As a CS grad, the first thing I saw was the pattern

nice!

I see you secretly deriving the geometric sum rule..