Perfect Number Proof - Numberphile
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- čas přidán 5. 01. 2015
- This video follows on from: • Perfect Numbers and Me...
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Objectivity: / objectivityvideos
Mersenne Primes and Perfect Numbers, featuring Matt Parker.
Matt is the author of Things to Make and Do in the Fourth Dimension. On Amazon US: bit.ly/Matt_4D_US Amazon UK: bit.ly/Matt_4D_UK Signed: bit.ly/Matt_Signed
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Is putting the lid on a pen the maths equivalent of dropping a mic, then?
Lol, first at 154 likes
@@rogervanbommel1086 I'm going for 155. : P
or throwing the last piece of chalk to the rubbish trash can
that’s called re-capping
It IS! (Caps pen)
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.
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."
It's not dramatic, or anything, just a quick stop / catch-breath type pause. Half a beat, so everyone knows what I'm doing.
i havent had much of this experience but i generally say "one less than 2 to the n" and "2 to one less than n" rather than having "minus one" in suitable place. others find it annoying for obvious reasons but i like it. :)
2 to the n ·*camera zooms in* minus one
me 2
to the n minus 1
Matt becomes the child he talks about at 3:43, at 13:37
A nice leet reference :)
Classic Mathematician:
"Let us assume that we know the total"
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
I got it for Christmas, it's brilliant!
Same, it was one of the best Christmas presents I ever got!
You make me want to go back to university! Why can't all teachers be like Matt?
Hey Brady, the videos in my subscription feed listed this one before (thus older) the previous one, which made it really hard to watch.. Just something to look out for. Really great videos nonetheless!
It's a really good read!
13:39 Literally the smuggest face ever :')
And admittedly: "...so pleased I'm going to put the caps back on both pens." ! ;-)
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!
+Ann Beckman
He tortures KIDS with it, not adults, whom I believe will see the pattern pretty much immediately :P
+Reydriel I'm 12 years old and I saw the pattern immediately... I'm also taking Geometry so I'm familiar with formal proofs already too.
+EpikCloiss37 12 year olds were doing geometry in the late 1800s, too. ☺️
Then what happened to our education system? Now you have to be in super special programs for that... (which are based on IQ of all things... Not a true measure in my opinion...)
I NOTICED THAT TOO WOW! :D
I USED A CALCULATOR TO DETERMINE THAT 8191 is multiplied by 4096 to get 33,550,336!
Oh that was beautiful; math truly is the music of logic!
Usually I hear people say the opposite, music is the math of art.
I agree!
Very well said... cool^^
+tggt00 Music is a massles body with a mathematical heart :)
Jasko Z Math is the science of the art of the music of logic.
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...
"I use this to torment young people" :)
3:27 Did he just call high school students "young"?
nice
@@quinn7894 secondary (high) schoolers start at around age 10/11 in Australia and UK (where he's from and where he lives respectively), which is young :)
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.
You should do more of these proof videos, this was really great!
Matt looks so happy at the end of this video :D
For some reason, I find Brady's incredulity at the beginning to be hilarious."You've already shown a link!"
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 just pronounce superscripts more quickly when they're together, like parentheses
Haha.. "that's why Australians are so good at math" 4:54
Matt is so funny
"Ethan, count to ten!"
"Yes, ma'am. One alligator, two alligator…"
(Yes, I know there are no alligators in the wild in Australia.)
I would have loved to see a proof of the other way around, that is every even perfect number has a Mersenne prime factor.
I think I would've gotten stuck at the geometric series step, but everything else was explained well and clicked for me. Cool!
So what about the proof that all (even) perfect numbers are of this form?
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...
Holy frick I am blown away, this is one of the coolest things ever
Love it! I may need to watch it again! Any plans for a Numberphile book?
Omg this would be amazing!
I saw you on tv! Outrageous acts of science!
Haha that's awesome.
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.
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.
Rising inflection,,, good work
I like this proof! Helped me to understand what was shown on the previous video.
13:34 turns to camera, looking very smug, "but now we've managed to prove it"...
"torment young people" LOL keep that up!
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.
Can't wait for objectivity! The onscreen links didn't work though, but the description wasn't far away :)
I saw the pattern, I've never felt so accomplished
I love these videos!
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)
Oh my goodness Brady is making a mausoleum channel.
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 havent even worked with binary I just learned that concept from a Khan Academy video showing how to count to 31 with your fingers xD. I feel like a special snowflake xD
TheRedstoneTaco
Or the binary number with only the nth digit =1 is exactly 2^n, 10000000.... -1 = n-1 ones, which is 2^ (n-1)
technically if you use both hands you could count up to over 1000 lol
and if you can do it with your thumbs they have 2 segments each ( some may say three including the connection to the wrist) and you get up past 1 million then.
Yes! I thought exactly that, the sum of powers up to n-1 is 1111111... with n-1 digits, and if you add 1, it becomes 1000... with a 1 and n-1 zeros, which is 2^n
love you, matt and brady
Isn't the pattern more clear in binary? Aren't we obscuring the pattern by thinking in decimal?
But to prove that about binary, you must still use geometric series, so in the end you get the same result either way.
Yes of course. This is far more easily understood in binary, so some of the algebra could be skipped, but the proof would still be necessary
6 -> 110
28 -> 11100
496 -> 111110000
8128 -> 1111111000000
The amount of 1s is n, the amount of 0s is n-1
"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 :)
Beautiful ❤️. Congratulations!!!
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.
Actually, Euclid proved this theorem and Euler proved its coverse (that all perfect numbers are of this form.)
@@leadnitrate2194
That’s… literally what he said…
@@KasabianFan44 I thought "one way by Euclid and by Euler in the other" meant that he was saying they proved the same thing two different ways, which isn't true.
Now that you're pointing it out though, I can see how I was probably wrong.
@@leadnitrate2194
Ahhhhh I see, my bad
Matt has got to be the best teacher ever.
Stand and deliver?
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)
***** lolwat. quantum mechanics is one of the most well empirically tested fields of physics there is. it has been thoroughly tested again and again and again during the entire 20th century. also, you'd be hard pressed to find any physicist at all who doesn't acknowledge it's validity.
***** are you taking about things like particle physics, super symmetry, super gravity, m-theory, super fluid vacuum theory, and loop quantum gravity. because they are not all subsets of quantum theory thought they use ideas from quantum mechanics they would be more accurately described as parts of theoretical physics.
I wish to be at one of his classes🤓
Superb video.
The perfect numbers are the triangular numbers of the Mersenne primes, or the factors that you multiply by are half the prime plus 1
Matt has a book. It's called " Things to Make and Do in the Fourth Dimension Parker Square". Check it out
Totally above my intelligence! Looking forward to next video
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.
I've seen Matt Parker in countless numbers of these videos and I just realized he reminds me of The Doctor.
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.
Is that proven, or have we just not disproven it?
@@Leyrann Euler proved it
An odd number can't be written in that form, and we don't know if there are any odd perfect numbers, therefore this isnt proven
@@coc235 they specified "even perfect number" so yes, it was proven
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
technically that could still mean (-1+2)^n, but i don't think any one normal would actually think that
+stickmandaninacan oh, yah. That hadn't occurred to me.
minus 1 plus the nth power of two is the only case that there's no ambiguity at all, i guess.
+stickmandaninacan No. (-1+2)^n is 1^n, which is 1. The order that you put the base and exponent matter with this operation.
Best IMHO is, "two to the n power minus one" vs "two to the n minus one power."
Completely unambiguous.
"to the" and "power" act like left and right parentheses there.
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?
The sum in the video is not even infinite.
Yeah, stuff can get a bit vague when you get to infinite sums. But this one's finite, so there's no real ambiguity to the result. The dots are not necessary, you could as well write the entire sum out and that way it's obvious all of the middle cancels out.
This method only works for infinite sequences whose sum converges. (Unless you're a physicist who doesn't care about rigour).
Wrong, infinity is a concept, not a number.
Well, you shouldn't because they're the ones that are right. That is a perfectly valid method for solving a finite sum, however it is COMPLETELY invalid for an infinite sum other than the small subset that completely converge. Using that method you can get literally any value answer you want. Look up the Riemann series theorem. It is well known that if you manipulate an infinite sum in this way you can arise at any solution you want. For instance 1+2+3+4+... can be shown using this method to equal -50, 2, 17, 99992, 1/6 and absolutely any other value (or also equally be shown not to equal anything).
More of this on numberphile would be appreciated :) this is maths
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.
Beautiful.
3:35 "professional jerk".
I'd love to see that as your profession on official documents.
Fun proof. Similar to a lot of the proofs I did when studying polygonal numbers.
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
Yeah! I found the Pattern for the Factors :D
I demand a Parker prime!
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.
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?
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
That was beautiful
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.
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!
Mathematics at its best
That's always fun. I remember being bored one day and trying to write a proof for the the quadratic equasion, I think it was nearly a decade before I found out what proofs were. So satisfying.
At 9:16, I start seeing two sequences multiplied by the Mersenne prime -- instead of just one.
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!"
I'm in high school and I got the pattern before you said it. I do feel smug :)
Me too. I'm also in high school.
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!
What I would give to have Matt Parker as my maths teacher...
The largest known perfect number, which is the 51st perfect number known, is (2^82589932)(2^82589933 - 1)
@9:09 a wrong factor (2^n-1) appears in the first line
Yeah
that hit me right in the maths
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!
I am in severe awe of this man's mathematical prowess.
About loking smug. Matt's look at the end.. :)
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
I realized the pattern had something to do with powers of two, it was actually the first thing I saw. I just hadn't worked out what they all were before he showed it, so I didn't get the chance to see the connection to the first column.
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.
As a CS grad, the first thing I saw was the pattern
im sorry, but what's a CS grad?
Matej božič Computer sciences graduate, i guess.
Slithereenn oh yeah.. probably, thanks!
Matej božič You're always welcome :)
congrats i saw it in less than 5 sec and i'm still an undergrad, anyone with a basic understanding of powers can see it stop gloating, in fact if a student can't see the pattern he should be worried
you dont need geometric series to solve that, just add 1 to the 1+2+4+..., you can see that the 1 you add merge the 1, equal 2, then 2 merge 2 equal 4 and so on until it is 2 to the n, and finally minus 1 which you added earlier.
I went through another day not having to use these calculations, again.
That smug face at the end! :D
If we use the equation (2^n -1)(2^n-1) I noticed that the number of divisors(including the number itself) of a perfect number is always equal 2n. Does someone know why is that?
A little-known fact is the converse of the theorem proved here is also true: If an even number is perfect, it must be of the form described here (i.e, 2 ^ (n - 1) * ((2 ^ n) - 1) ). This was proved by either Euler or Fermat, I'm not sure which. The proof is also longer than this one.
I found the 2,4,16,64 incredibly quickly. I’m not a genius, I just had already read the top comment
Excellent presentation of the topics. Many many thanks. DrRahul Rohtak India
Interestingly enough, one way to tackle the 1 + 2^1 + 2^2 + … 2^(n-2) + 2^(n-1) summation is to write it in binary. What happens when you do that is you get a binary number that’s a series of 1s that’s n-1 digits long, so if you’re familiar with how binary numbers work it becomes immediately obvious what the sum is.
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!
Pattern at the start seemed obvious to me, but a interesting video none the less
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) + .........
9+10=21
Agreed, same mistake at 9:19
I think that was originally supposed to be a reminder, that the sum in that line adds up to (2^n)-1 ... but using commentary with round brackets in equations is not a smart thing to do.
***** or 9+4=30
yes, plz correct, i try to follow along but mistakes like these can literally throw the video out of wack
Can you prove this problem via Induction of the series?