Catalan's Conjecture - Numberphile
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- čas přidán 13. 02. 2018
- With Dr Holly Krieger from Murray Edwards College, University of Cambridge.
Have a look at Brilliant (and get 20% off) here: brilliant.org/Numberphile
More links & stuff in full description below ↓↓↓
More Numberphile videos with Dr Krieger: bit.ly/HollyKrieger
Her Twitter: / hollykrieger
Some more reading on the topic: www.ams.org/journals/bull/2004...
Open Problems Group on Brilliant: brilliant.org/groups/open-pro...
Editing and animation by Pete McPartlan
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
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I have purchased lots of brown paper and magic markers but I am still useless at Maths.
Marcus Campbell Yes I know it is corny
You need sharpies
John Ayres
...but you’re very fashionable while being useless. A Kardashian of maths.
dumbass
*Dr Holly Krieger is so white and redhead that i need my dark glasses to even see* . 😂😂😂😂😂😂😂😂😂😂😂😂
For an elder (+60), average electronic engineer with a major interest in math, this channel is awesome.
I'm currently studying electrical and computer engineering. Has your job been fulfilling?
Yeah, of course it has. He's so full he can't even move.
I know its you electroboom @electroboom
I'm 21, Been Watching since 17 :D
uh, you make me feel old.. im just 17
In my first semester at the Georg-August university in Göttingen (Germany) the linear algebra lecture was given by Preda Mihailescu. Nice to hear his name in one of our videos!
That's awesome!
My conational😎
That course is still infamous at Göttingen uni as the "linear algebra course which almost nobody passed" :D
@@TheMrbaummann I passed it in 2019! Preda is totally awesome
Göttingen needed such a mathematician after Hilbert, Dirichlez and Gauss.
It is more fun to write the equation as 3^2-2^3=1^23
Or 3^2-2^3=3-2
Dale Kerr Quite eXcellent !!!
Do you mean 3²-2³=1²³?
:D
@@pablozumaran3997 HOW
@@MatBaconMC Key combinations: AltGr+2, AlgGr+3.
My 9th grade math teacher called perfect powers “sexy numbers”
Sounds inappropriate tbh
@Diego Maradonna although those would be called sexy primes
@@nexusclarum8000 You sound pointless tbh
My 9th grade math teacher called me Nick-mobile, then I found out he called Steve, Steve-mobile. I was devastated, though I was special, guess not
As they should be
I do love that moment of clarity and understanding when learning something new (...I also enjoy watching someone else experience it when I am the one teaching). Most of the math in these videos goes over my head, but I always seem to get just enough to get a brief moment of learning. Thanks again for doing these videos!
when you're sitting alone on Valentine's day and numberphile makes a new video.
Thank you numberphile, atleast you give me math.
😂👏🏼
And Holly.
26 is the only number that simultaneously is one more than a square and one less than a cube.
Can you prove it?
Maxi Lexow Yes, it uses unique factorization in Z[sqrt(-2)].
now its known as john cessant conjecture
This problem currently appears in brilliant advanced weekly problem.
Yes so is a unique soln to x^2 + 1 = y^3 - 1. (x,y) = (5,3)
Catalan's Conjecture is too strong a theory and wants to separate from the rest of mathematics. It wants to be in its own independent set. Can't blame it.
[Catalonia joke]
There is nothing in the mathematics constitution that allows this conjecture to separate itself.
lol catalonia
Asturias> Catalonia > rest of Spain > rest of Arab blood filled nations.
Damn your racism is over 9000.
Get back to your mine.
OMG a new conjecture of math!
"This conjecture was already proven"
WHY DON'T CHANGE IT TO A THEOREM????
Eduardo Muller
It's been proven by Mihailescu
You can legally call it Mihailescu's Theorem
Alliteration. The only reason
I would think that it is because it's probably an old conjecture, so people are just used to calling it and referring to it as a conjecture?
It was Catalan's Conjecture. There is no inconsistency in this terminology.
Just to sound very very very very very very tough
This Conjecture was proven by Preda Mihăilescu, at the University of Paderborn ! Honored to be able study at the university in 1 month's time ! XD
He is teaching now in Göttingen, you would have even had him in linalg 1&2 and algebra if you started 2 years ago in Göttingen.
so how did it go? hopefully you learned a thing or two!
I love paderborn
@@goldminer754 I had him in my AGLA1 course. He is a really cool guy, his lecture is a bit all over the place though. Proving the fundamental theorem of Algebra to first semesters the Gauss way isnt really cool. Möbius transformations arent nice either for 1st semester students!
Let's go romania.
That was really fun. I think it is nice to go ahead and start down the right path....even if we can't follow the whole big proof.
Thanks!
2:56 I'm kinda surprised that this was proved algebraically. Most difficult number theory problems seem to be tackled analytically nowadays.
If you look closer it's about groups, space and abstract algebra. How would you prove otherwise
Bringing Holly in was the best thing numberphile has ever done
That's not fair to Hannah Fry!
Of course, there's always a next question(s), once something like this gets settled. Like, is there a point beyond which there are no more differences as small as d, where d is 2 or 3 or ...
For instance, are 25 and 27 the last pair of powers that differ by 2?
Are 125 and 128 the last pair that differ by 3?
Are 2187 and 2197 the last pair of powers that differ by 10?
Etc.
Thanks! This was fun!!
Fred
PS. A reply 2 years ago, by dlevi67, to a similar comment of mine, points out that, "Pillai's conjecture says that there are only finitely many misses for any integer value of the miss."
Great video as always. Icing on the cake was the Public Service Broadcasting LP in the background. Excellent choice!
when you’re single and have to watch math videos
WANT and CHOOSE, not "have to", pffff
Dont worry. I'm engaged and I'm still watching math (and some history) videos. ^^ Math loves you!
and you don't even study mathematics
XSimoniX so true bro
😂
I don’t “crush on” CZcams celebs but omg I think I’m in love.
I've never found any category of people to be categorically excluded from crush potential. Patterns of people are about as useful as patterns in clouds.
Bob Stein
He's propably saying that he's not a tennager who loves somebody just because he like the videos they make. The "pattern" can inform about the people he like or why he likes them. And patterns of people are totally useful. We classify people all the time because of that.
Bob Stein What about Trump supporters?
I have a thing for women that show a passion and enthusiasm for something!
I could listen to Holly explaining anything all day and not mind at all, some people have that special something.
What I love about this channel is shows that math can be super hard but at the same time doesn’t require you to be from a different planet to understand it.
This channel is great! Thank you for another great video.
I love Numberphile videos featuring Dr Krieger!
...Happy Valentine's Day y'all, I guess?
Great video. Hopefully you'll do more proofs with Holly
Nicely done! Making math entertaining as always!
Love the mighty Nail and Gear in the background!
Numberphile needs some t-shirts and other merch, man. So many cool things you guys cover.
Please do more with Holly!
Quarantined sitting on toilet and watching Dr Holly’s videos. This could last for months.
Mihailescu is currently my professor for linear algebra. A very kind and jolly man.
Nail and Gear picture in the background! Nice crossover!
They grey right?
Very interesting problem. Happy Valentine's Day!
I just started learning about elliptic curves, and the curve y^2=x^3+1 is an elliptic curve of rank 0 with a torsion group of order 6. Not only are there no integer solutions (other than (-1,0), (0,+/-1), (2,+/-3)) but there are also no other rational solutions!
you make mathematical concepts intresting. thank you
I've been puzzled by 2 cubes in geometry in recent time, would you provide me with your interpretation, please?
Dr. Holly Krieger is perfect for a Valentine's Day Numberphile!
You're a weirdo.
What
@@lincolnsand5127 lol
Dr. Krieger seems like she'd be just so much fun to hang out with!
I can't shake the impression that Dr Krieger reminds me of Jewel Staite (Kaylee in Firefly, Dr Keller in Stargate Atlantis). There's something about the voice that rings the same bells, as well as the way she looks when she smiles.
Thumbs up for the Romanian mathematician !!!
I thought he was Catalan...
A noastră!!
@@sharoneisenberg2274 the mathematician who proved the conjecture is Romanian
@LookBackTime
Nice to have a famous Romanian other than Count Dracula.
I'll get my coat.
Theoretically, vampires are intelligent beings
so really charmming!!
Perfect guest for Valentine's day video.
I'm interested to know where the difficulty arises from when there are equations involving both addition and multiplication.
Preda Mihāilescu.........what!?I can't believe that a romanian made it to numberphile I am so proud 🇷🇴🇲🇩🇷🇴
Dr Holly Krieger😍😍😍😍
claiming Hannah Fry
RIGHT?!?
*Dr Holly Krieger is so white and redhead that i need my dark glasses to even see* . 😂😂😂😂😂😂😂😂😂😂😂😂
This boss level beauty
Well, it's nice to see that frame with Ron Graham's handwriting in the background.
Excelente video. Los felicito!!!
She's intelligent and incredibly charming. What a perfect combo.
😍
beautiful beautiful
Beauty with brains
Dr. Krieger, you are a true unicorn :)
Holly Krieger is back 💙💙💙
One step of the proof reminded me of a PBS video about Sohr's algorithm. Intriguing
Of course you got Dr Krieger for valentines...
I ain't even mad tho.
I've got the feeling that, all of a sudden, a lot of people are going to become very interested in maths.
Best kind of valentine's gift! Niiicee
Into to Algebra 1. Best explanation I’ve ever heard. 👍
here I am, laying in my bed watching youtube videos, not learning for my exam tomorrow and there is a video about one of my professors at the university of Göttingen 😂
3:09 - "We don't have time for the next 'couple of years'..." - *Looks at watch* - I thought that was very funny.
I MISSED DR KRIEGER SO MUCH
Brilliantly explained
What I think is even more interesting about those numbers is: Is every natural number a difference between two of those Catalan numbers?
Nobody's proved it for the number 6, let alone "every natural number." Or for 14, or 32, or 42, or 50... there's an apparently infinite number of (conjectured) counter-examples (A074981 in the OEIS)
Came for the mathematics, stayed for the mathematician.
Preda Mihăilescu is Romanian. I was so happy to see his name in this video.
Greetings from Romania! Cheers to Preda Mihăilescu😙
EVERY FREAKING TIME I look at the title I think of Catalonia
Oooh, a public service broadcasting's race for space! A great album.
I love the nails & gears flag in the background in the beginning ;)
Dr Holy Krieger ❤️❤️ Long time no see ❤️❤️
2:50 Wait, that's a French poem, not math!
OMG a fellow romanian demonstrated this? Nice one.
Yes, a Romanian
Possibly worth mentioning that if you extend to non-positive integers, you also get (-)1^n and 0^n (but no other pairs separated by 1 - negative integers are only powers if they're odd powers of negative integers). Possibly not worth mentioning it either.
Is that a public service broadcasting vinyl I see in the background ??
*Studying conjectures is my passion.*
I conjecture you have yet to find your life's most interesting conjecture.
(Unless that was it. But then it was still true when I conjectured it.)
*"Yeah that's exactly right"*
I know the mathematician who proved Catalan conjecture. Prof Mihailescu did it in 2004 and gives lectures in Göttingen where I used to study math.
Thanks for the link to complete proof.
Isn't that the *CGP Grey logo* standing in the corner? :D
that's a funny way to call it!
I think it is. Perhaps it's subliminal cross-promotion.
It's the Nail and Gear! Flag of the Hello Internet podcast.
"CGP Grey" is a funny way to spell "hello internet"
Where is numberphile live from maths fest?
so does it work for all separations? Is each separation unique?
You're awesome mate!!!
Super awesome video. Question, are there not two cubes that are two integers apart? That being -1^3 and 1^3, and their results would be -1 and 1, respectively. Need to rewatch video as often this stuff goes over my head the first time.
There seems to be an unstated assumption here. At around 5:30 we're told there are no two cubes that differ by two. But there is one such pair: 1 and -1. Thus, we can expect a solution if x = 0. And putting that into the original equation gives us y = -1.
Is 1 the only difference that only occurs once? Or the only number that occurs a finite amount of times for that matter?
Given the Brilliant problem in the end, I suggest taking your Pick of the answers.
An interesting observation I've often wondered about, but had no idea was actually being tackled by mathematicians!
There are a number of other "pretty-close" cases.
5³ - 11² = 125 - 121 = 4 · · · ↓
2⁷ - 5³ = 128 - 125 = 3 . . → these two examples are all the more interesting, because there are *three* powers within a short span (7)
13³ - 3⁷ = 2197 - 2187 = 10
Pillai's conjecture says that there are only finitely many misses for any integer value of the miss.
And 2209 = 47^2 comes shortly after 2187 and 2197 too, so there's another bunched up trio of powers
I literally typed in “that conjecture where there are only two palindromic powers and they do some things”.
Super awesome!!! Was enjoyable!! Hugs! ❤️🚀🦟🎶
Is there any case of n^m = m^n other than 2 and 4? Been curious for a while
It.s ROUMANIAN and we did not know about him!
Ikr
thumbs up for the romanian mathematician
Are you Romanian?
yes
Preda Mihăilescu for President, he's currently hibilitated in göttingen, Germany which is where i am studying
Sal fra
My fav math prof is Romanian! And one of my fav ow players. Start to develop a strange fondness for Romanian people :p
Funny how the most complex of problems could be turned to something so simple , with just a little bit of creativity :)
Nice video!
Are there an infinity of perfects powers separated by 2? and 3?
Pillai's conjecture is that there are only finite numbers of perfect powers separated by any integer value. Still not proven rigorously, as far as I know.
dlevi67 thanks that's fascinating, i'll check this out
Pillai's conjecture states that each positive integer occurs only finitely many times as a difference of perfect powers. That's a less ambiguous way of stating it.
I fail to see the ambiguity (or the difference - finite or otherwise), especially considering the way in which the question was formulated, but if it makes you happy... ;-)
YASSSSSSSSSSSSSS I LOVE CATALAN'S CONJECTURE. Something about it is just so amazing.
ha, nurd
it's the green eyes and the smile of the conjecture
I enjoy these videos
I like how I failed every single aspect of math throughout many years of schooling and yet somehow by watching this video I naively thought "oh hey, you're older now Trevor, you'll probably understand what's being said"
If
"for any n, there exist perfect powers differing by n"
hasn't already been conjectured, I demand that it be named after me.
Ah, the Jay-Mu-Doc conjecture
I feel like the larger n is, the more examples of perfect powers differ exactly by n. The problem, in the form of "a^b - c^d = n" is far too free to have any integer n that doesn't also have at least one set of integers a, b, c, and d
I think it's conjectured that there are no two perfect powers differing by 6.
I am a complete amateur in this field but -1 and 1 are cubes and they differ by 2, so why not use them. Or is it just that we are only thinking about positive integers only and not the negative one?
the conjecture says : x^a-y^b=1 with x,a,y,b > 1
Powers start with higher power of 2. Then power ordered. A square minus cube is one. Folding geometry is unit. 2 and 3 are the closest in switch over jumping into negative dimensions.
Odd question but, what's that music album that is in the background, if anyone knows and if it's an album, I think I saw it somewhere.
(a+b)^2 st!
Of coures they are the same area - see Pick's Theorem
I wondered why Beady didn't remember the Numberphile video about Pick's theorem, then remembered it wasn't a Numberphile video I was remembering.
I've read that 26 is the only integer that falls directly between a square (25) and a cube (27), and that Fermat proved it? Is this right and is the proof similar to this?
May I propose the Schultz Conjecture: looking at the list of all integer powers, a separation of every integer value is found eventually, like there is *somewhere* a separation of 1, 2, 3, etc.