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United States
Приєднався 18 сер 2024
Howdy! This is a channel I've made to showcase interesting ideas in the fields of mathematics, computer science, physics, etc. so that others can see hard-to-understand concepts in an intuitive way.
I'll probably upload monthly, but since I'm a college student, there will be times where my upload schedule is a bit infrequent.
I'll probably upload monthly, but since I'm a college student, there will be times where my upload schedule is a bit infrequent.
2025 will be an interesting year . . . (Nichomachus's Theorem)
2025 is an interesting year for a few reasons. Most notably is that it is 45 squared. It's also a rare number that satisfies Nichomachus's Theorem, which is an amazing theorem with a beautiful proof.
Reupload due to an algebraic mistake.
00:00 - Introduction
1:16 - Algebraic Proof
3:45 - Geometric Proof
5:36 - Conclusion
Music Used (With Permission)
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Real Book Solo Piano - Heritage Hum (Jimmy Heath) - Performed by Budo Saurus
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You can find other performances by Budo Saurus here:
www.youtube.com/@BudoSaurus/featured
Reupload due to an algebraic mistake.
00:00 - Introduction
1:16 - Algebraic Proof
3:45 - Geometric Proof
5:36 - Conclusion
Music Used (With Permission)
---------------------------------------------------------------------------------------------------------------------------
Real Book Solo Piano - Heritage Hum (Jimmy Heath) - Performed by Budo Saurus
---------------------------------------------------------------------------------------------------------------------------
You can find other performances by Budo Saurus here:
www.youtube.com/@BudoSaurus/featured
Переглядів: 3 033
Відео
This Theorem Has a One-Sentence Proof (Fermat's Christmas/Two-Squares Theorem)
Переглядів 22 тис.21 день тому
Exactly 384 years ago today, Pierre de Fermat would write a letter showcasing one of the most important theorems in number theory. A few centuries later, a man named Don Zagier showed the world one of the most beautiful proofs of the theorem, using only one line! 00:00 - Introduction 1:11 - Windmills 5:10 - Involutions 7:47 - Zagier Map 10:20 - Conclusion Music Used (With Permission) Christmas ...
Paper Hats, Proof Techniques, and the Jump into Pure Mathematics #SoMEpi #SoMEπ
Переглядів 5615 місяців тому
This is just a simple video I made to introduce people to higher level mathematics, since it's quite different from the math you know in high school. Music Used (With Permission) Real Book Solo Piano - Dolphin Dance (Herbie Hancock) Real Book Solo Piano - Bewitched, Bothered, and Bewildered (Richard Rodgers and Lorenz Hart) Performed by Budo Saurus at www.youtube.com/@BudoSaurus/featured
Inductive geometric proof assissted by algebra: the side length of the square is given by the sum up to n, which is n(n+1)/2. We wrap the new cube area around two edges and a corner of this, which means we cover n(n+1) units of area multiplied by the thickness t, plus t^2. if t=n+1 then we get n(n+1)^2 + (n+1)^2, which is (n+1)^3, showing that the next number cubed perfectly marks the increased area of the square, while growing the side length by n+1.
🥱yawn.
Loved this!
It's not at all clear to me that a general cube would lay out in just the right L shape. I guess i can revert to algebra. The corner is n^2. Each leg is width n and height sum of numbers to n-1, so add them all up: n^2 + 2 * [ n * (n(n-1)/2) ] = n^2 + n^3 - n^2
Hi😊
A thing thats cool is that 2025=45²=(20+25)²=(1+2+...+9)² This will also happen in 3025 (as said in the tumbnail) 3025=55²=(30+25)²=(1+2+...+10)² 🤯🤯
well that waas really cool, props to you man
8:27 gahhh... rotate the squares.
also, for the earlier rectangles, u could just keep rotating the same direction, and attach other end back, since it's not like there's elbow joint
You're criminally underated, man. Keep up the good work
Came here for nichomachus's theorem ❌ Came here for the soothing voice ✅
fun fact about 2024: 2024 is actually a tetrahedral number meaning its a sum of the first n triangle numbers, or a sum of the sums of the first n natural numbers. (so like (1)+(1+2)+(1+2+3)+(1+2+3+4)+...)
Math Olympiad competitors: write that down!
Today, every way to write the date contains only square numbers
Well... Writing the date in the Julian calendar or the Hebrew calendar or the Chinese calendar are also ways to write today's date
Interestingly, 2024 can also be written as the sum of cubes, namely starting from -1 up to 9. However, I am pretty sure that Nichomachus's theorem fails when starting at a negative number.
Yeah, I think it has to start at 1. (Or 0 technically but that’s trivial)
Here is a new subscriber, well-earned! ALSO, cats are nice!
Fun video !
At 3:05 we can also note that both terms have a factor of (k+1)^2, which makes the numerator equal to (k^2 + 4k + 4)(k+1)^2 = (k+2)^2 (k+1)^2. makes the factoring a bit easier. love your videos btw! :D
Oh wow, didn't even realize! I ended up factoring the hard way. And thanks for the support, it means a lot!
@Zscore6174 as a rule of thumbs i always factor before expanding.
Another great video from the goat 🗣️
A short and simple video for a short and simple topic. 0:24 was a super cute touch.
You are going to go viral one day, remember us
Agreed. Remember us OGs.
Agreed
Agreeed.
i waanna be remembered
Yeah
Neat.
AI can proofread better than bro
AI can also come up with more creative and factually based insults...
You should master your recording (or get a better mic), it's very harsh. Also, there's more focus on the vocabulary than I think is really justified. I kept expecting you to actually show the set of possible transformations for specific involutions and pair the swaps together to show the point you're making, but you never actually did the second thing. I feel like your visual explanation missed showing how nonprimes have more than one fixed point, and how that interferes with the proof; you could easily have shown an example of that and it would have cleared up a lot.
This Theorem Has a One-Sentence Proof. Q.E.D.
could you please explain the value of this proof or theorem to number theory? I don’t see it as particularly generative of other results or of practical importance. I have studied some number theory and seeing the Christmas theorem before. You provide quotes or blurbs endorsing its importance, but can you explain its importance? This is not a sarcastic comment. I earnestly want to know why this is important and what am I missing? I like your video so far. I think you just need to better motivate the importance of the theorem.
could you please explain the value of this proof or theorem to number theory? I don’t see it as particularly generative of other results or of practical importance. I have studied some number theory and seeing the Christmas theorem before. You provide quotes or blurbs endorsing its importance, but can you explain its importance? This is not a sarcastic comment. I earnestly want to know why this is important and what am I missing?
Certainly. Probably the best use case of Fermat's Christmas Theorem is in studying binary quadratic forms, which are equations of the form q(x,y) = ax^2 + bxy + cy^2, which are useful in studying Diophantine equations, elliptic curves, etc. It should be clear that Fermat's Two Squares theorem is a special case of a binary quadratic form, namely when a is 1, b is 0, and c is 1. In general, I've found that number theory by itself doesn't appear to be useful, but the results of number theory are applicable to other fields of mathematics. A really good example of this is the video "Pi hiding in prime regularities" by 3Blue1Brown (ua-cam.com/video/NaL_Cb42WyY/v-deo.html) since it uses Fermat's Christmas Theorem in the context of Gaussian integers.
@ wow! Thanks very much.
So coincident that I've just read the 2 squares theorem on a number theory book this week after I read the chapter about quadratic reciprocity. Never wonder there is such a simple and mind-blowing alternative proof! This theorem is also a question left in a video of 3Blue1Brown ( ua-cam.com/video/NaL_Cb42WyY/v-deo.html ) about decomposition of Gaussian intergers and a formula for π derived from it. But I also have another question about the proof in that video and hope if you can explain, that why the arithmetical function χ(n) appear out of thin air (at about 20:00 in the video) and instantly kill the problem? 3Blue1Brown first explained how to count Gaussian integers of absolute value R, and then smartly and unexpectedly use the χ function to rewrite the formula by treating all the prime numbers equally. After a little simplification, you get 4 times of χ*u (where * is Dirichlet convolution and u(n) = 1 'cause I've learned a little analytic number theory) !? That's surprising but unmotivated. Can you give a reason why χ appears? Thanks!
11:16 Agree that math is just soooooo beautiful that I shall spend a lifetime learning and appreciating math!
Looking forward to your new videos!
10:46 I know that there is exactly one solution to each prime number p of the type 4k+1 to be represented as the sum of 2 squares. But is there an intuitive way to understand why this happens, probably in the language of windmill?
I cant hear you over the piano.
@StylishHobo Yeah, I know the audio quality is meh, but no worries! Future videos won't have this problem. I'm mostly leaving this one up since I did it for a math competition, and I feel it'd be a bit against the spirit of the rules for me to a reupload, especially four months after the fact.
No comment, just like and subscribe, and of course sharing this elegant piece of math to my friends Good luck In upgrading math content in youtube!
I was here before 1,000 subs 🥳
Holy goated math video!!!
incredible proof and incredible video.
My Respect
"Alright, we explained what windmills and involutions are, now we just have to combine the concepts." "Well, the first 7 minutes of the video was easy, so the rest shouldn't be too hard, right?" "..." "Right...?"
Didn't Mathologer do a similar video?
Great video, like others said I would maybe slow down the pace a bit and explain the equations more with more dynamic visuals
bro's thumbnail almost become rejected from art school
This is very intuitive and fun to know about conversion of a mathematical statement into a nice windmill
Less than 1k subs.... for now. This is incredible man. Dont stop making content u rock
Best animation based on math and geometric proof
thank you math kitty for this blessed video
>sees thumbnail *sigh i can smell the comments
But come on breaking up B into y and z is totslly contrived and nonone wouldnever think of thst sonwhy not do without thst step?? It's an unfair cheat don't you agree? No one is goign tonthink of thst unless you know the answer beforehand..and does anyone know how Fermat discovered in the first place all lrimes of tbis form can be written as ssuka lf two squares?
nice, interesting video, your voice is so calming...☺️ also i like your cat character 😄
Can you get a similar result in 3d?
There are a few, yes. For a similar result in 3d, you'd have to consider summing cubes. There's a few various theorems related to summing cubes, such as Nichomachus's theorem, but I think there are 3 theorems that are closest to Fermat's Christmas Theorem. - The first is something known as Waring's problem. Essentially, we know that every natural number can be written as the sum of four cubes (Lagrange's four square theorem) or the sum of 9 cubes, but there isn't really a special case with prime numbers and cubes. - There is an open problem in math, however, called Sums of three cubes. It asks "Is there a number that is not 4 or 5 modulo 9 (that is, when you divide the number by 9, you get a remainder of 4 or 5) that cannot be expressed as a sum of three cubes?" - There's also another unsolved problem in math known as the Sum of four cubes problem which asks if every integer (positive and negative) can be represented as the sum of 4 perfect cubes. In terms of a similar visual result (aka a proof using windmills), it's a bit difficult since proving Waring's Problem on cubes is a tad bit more complicated than Fermat's Two Squares Theorem. And since the other two are unsolved problems, we don't know if there is a nice geometric interpretation, let alone if there is any proof.
Good video but it wasn't clear how the "trivial" solution where x=y is the only fixed point of the Zagier Map. I can see it visually though
OK, I’m not versed on mathematics, but sometimes interested in topics. To me, this video goes way too fast, juggling with all kinds of x’s and y’s without quite knowing what they stand for makes it impossible for me to get what you like me to understand. OK, let’s share the misery: I’m too stupid, you’re too fast.
great video :3
Please do make more videos this was great :)
New sub!!