Is this the most beautiful proof? (Fermat's Two Squares)

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A visual explanation of one-sentence proof of Fermat's Two Squares Theorem by Don Zagier.
Support me on Patreon! / vcubingx
This video adapted a lot of material from this paper: arxiv.org/abs/2112.02556
If you're interested in this proof, I highly recommend it and Proofs From The Book! (In general, if you're interested in math, you should read Proofs From The Book)
Some Clarifications:
At 9:00, I meant to say 3^2 + 2^2. Sorry :p
When it comes to the Zagier Map, one question a friend asked is that we only ever consider cases where the inner square is the smallest square or the largest square - what about the in between cases? To this, try and think of if you can create a windmill with an "in-between" case. If you can/cannot, why? (spoiler you can't!)
The source code for the animations can be found here:
github.com/vivek3141/videos
A huge thanks to Chaeyeon Lee, Youngjin Park, Oliver Ni, Eric Che and Antonio Kam for helping me review the video!
These animation in this video was made using 3blue1brown's library, manim:
github.com/3b1b/manim
Chapters:
0:00 Introduction
1:40 Part 1: The Involution
5:42 Part 2: The Windmill
10:47 Part 3: The Zagier Map
15:12 Putting it all together
17:13 Outro
Follow me!
Website: vcubingx.com
Twitter: / vcubingx
Github: github.com/vivek3141
Instagram: / vcubingx
Patreon: / vcubingx
Music (in order):
Philanthrope, mommy - embrace chll.to/7e941f72
GameChops - National Park
Philanthrope, Idealism - Still chll.to/a110849c
GameChops - Route 113
Helynt - Route 47
Helynt - Bo-omb Battlefield
Helynt - Undewater
Knowmadic - Faces chll.to/892bc12e
Helynt - Littleroot Town
Hope you guys enjoyed the music! I especially had fun picking out the tracks for this video. S/o to Helynt, they make amazing remixes!
The most beautiful proof in math? (Fermat's Two Squares Theorem)
Some tags: number theory, fermat's two squares, windmills, proofs, mathematics, vcubingx v cubingx, vcubing x

Опубликовано:

1 июл 2024

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Комментарии : 161

@vcubingx 6 месяцев назад

Hi all! It's been a while since I made a video. Was busy with work/school stuff, but now that that's wrapping up (hopefully!) I should be making more videos! I had a ton of fun making this one, especially picking out the song choices. Let me know what you think of this one!

@mathiarasi8012 6 месяцев назад

Good to see u again Buddy 😊🎉

@guillaume5313 6 месяцев назад

Great video ! There's a small mistake though at 8:55, you say 13 = 3^2 + 4^2 instead of 3^2 + 2^2

@BigAsciiHappyStar 6 месяцев назад

is that the Mario World tune in 3/4 time? 😃

@gurmeet0108 6 месяцев назад

That was a very well done video!!! I remember Mathologer video on the proof, it was pretty good too, you might like to add that in description. - ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-DjI1NICfjOk.html . And also you might like to add that the geometric interpretation that you mentioned was given by Alexander Spivak as far as we know, as mentioned in the description of the Mathologer's vid.

@eylonshachmon6500 6 месяцев назад

The fix your put in the description says “13 is equal to 5^2 + 2^2” instead of “3^2 + 2^2”

@lowellrindler9454 6 месяцев назад

“My favorite thing about number theory is how simple the theorems are to state but how wildly complicated the proofs are.” This is also what I find so maddening about number theory, but to each their own haha.

@table5584 6 месяцев назад

I proved the collatz conjecture. Where do I submit my proof?

@melone3631 6 месяцев назад

@@table5584 calling cap but if you did, explain

@table5584 6 месяцев назад

@@melone3631 Basically, I tried to determine a function (let’s call it g(x)) that is bijective, and that’s also increasing to inf, and that also follows a similar modified version of “the 3x+1 and x/2” function that tries to find the number of p^n less than x in g(x), and using some category theory, and algebraic topology (with some other stuff), I discovered that there was no function, meaning that every natural number inputted enough times into “the 3x+1 and x/2” function eventually and inevitably goes to 1. For now, I am gathering all the papers that I have used to solve the proof. There’s a couple hundred papers on my desk that I will have to review for any mistakes. I won’t submit my proof yet until I review and revise it a couple of times to make sure my proof is correct.

@melone3631 6 месяцев назад

@@table5584 Heavy respect, your name will go down in history if it proves the conjecture. Happy to be there when it was written :D I'm waiting for the papers!

@eta2321 6 месяцев назад

@@table5584damn thats sick, im very skeptical but if it ends up being correct i hope you can get your proof published

@rahulshah6119 6 месяцев назад

At 8:56 i think "this tells us that 13 is 3^2 + 4^2" should've been 13 = 3^2 + 2^2 instead (in case anyone else was caught up by that), great video and incredible usage of graphics in a math proof!

@vcubingx 6 месяцев назад

Thanks! Yeah the mistakes on me…missed it during editing

@paulbloemen7256 6 месяцев назад

I am not a mathematician, but sometimes interested in patterns. I find this proof, and the subject itself, with all its aspects like windmills, totally mind boggling, I am so surprised that something like this even exists. Thank you for showing, and patiently explaining!

@artemetra3262 6 месяцев назад

omg. that's actually so elegant - one of those surprisingly common cases in math when a problem is extended to involve more variables and general statements in order to then fall back to the ground because some random pattern is seen.. these proofs are the ones that impress me the most, i have no clue how does one even come up with those things haha also, you just gained a subscriber!

@vcubingx 6 месяцев назад

Thank you! Yeah me neither...I have no idea how someone could think of this.

@user-wc5re2ml9j 6 месяцев назад

Fermat's Two Squares Theorem generally also includes the statement that the representation as a sum of two squares is actually *unique* for such primes. I don't think this proof can easily be extended to show this.

@vcubingx 6 месяцев назад

It most definitely can! See the attached paper arxiv.org/abs/2112.02556

@user-wc5re2ml9j 6 месяцев назад

@@vcubingx That paper *mentions* that there is uniqueness (in it's Theorem 4.3), but does not actually prove it. It does add that the proof is "purely number-theoretic" and, moreover, that the proof has been formalized by Laurent Théry. The cited proof doesn't seem to use this windmill formulation (does it?). I *like* the windmill formulation and proof, but again it seems to me that the uniqueness needs to be shown separately by a different technique.

@vcubingx 6 месяцев назад

@@user-wc5re2ml9j You're right! I missed that.

@yourgrandfather3169 6 месяцев назад

The notion of fixed point is so amazing. For a set S, if we can prove that there is an involution f, such that f has odd number of its fixed points in S, then every involution on S has odd number of fixed points on S either. Here we can prove that Zagier map is an involution and has (1, 1, k) as an only 1 fixed point on W_(p=4k+1z). Because 1 is odd, so all involution on W_(p=4k+1) including the flip map has at least 1 fixed point on W_(p=4k+1). Since having a fixed point of the flip map on W_(p=4k+1) implies that p=4k+1 can be written as a sum of two squares, therfore we now have proved the theorem !

@aaronmartens2903 6 месяцев назад

Pretty slick, thank you. Counting things in two different ways is super powerful.

@mavericktron2966 6 месяцев назад

For a 4k+1 prime you start with the (1,1,k) windmill put a pin in the middle and reel it in counterclockwise (aka involutions) then the middle gets fatter (Zair map) and you do it until you end up with four nice square flippers at the outside ... yah probably not exatly like that but that's how I saw it anyway 🙂

@jupa7166 6 месяцев назад

I'm to dumb for this but You've explained it beautifully. Well done. Who comes up with proofs like this? Insane.

@djridoo 6 месяцев назад

Pokemon and Mario music on beautiful and elegant maths, what a cool video !

@shivbhatia2784 6 месяцев назад

This was an incredibly clear explanation, amazing work!

@vcubingx 6 месяцев назад

Thank you!!

@leif1075 6 месяцев назад

@@vcubingxwhy di you like thst it is so complicated don't you find thst infuriating or frustrating?

@Rollinghills1 6 месяцев назад

Here’s my one line proof. “This is left as an exercise for the reader.” 🔥🔥🔥

@GrandMoffTarkinsTeaDispenser 6 месяцев назад

Congratulations on this beautiful video. A joy to watch from start to finish. This topic is very dear to me.

@-tsvk- 6 месяцев назад

The "windmill" construction is about warpping a 2d square with 2d rectangles on its all four sides, visualizing a mathemathical property/identity of the form a^2 + 4*b*c Does there exist a corresponding construction in one dimension higher, wrapping a 3d cube with 3d cuboids on all of its six sides? Visualizing some possible related identity of the form a^3 + 6*b*c*d ?

@nurulhuda2123 6 месяцев назад

Happy to see you back!

@varunmohanraj9499 6 месяцев назад

Omg the goat returns my favorite youtuber ❤️❤️❤️

@vcubingx 6 месяцев назад

Holy it’s the goat im truly honored

@glassfish7207 6 месяцев назад

Always enjoy your math videos with that elegant animations! You always remind me of the beauty of math!❤🎉🎉

@vcubingx 6 месяцев назад

Thank you!! Hope to make more videos :)

@leif1075 6 месяцев назад

@@vcubingxjust to clarify, you don't mean that y HAS to equal z for this to be true right? There could be primes of thus form where you and z are different and you form the two squared terms in different ways like with the other tern not being x squared right? I don't see why not. Though don't know if those are rare. Thanks and hope to hear from you.

@chiaracoetzee 5 месяцев назад

This is a brilliant and really accessible explanation of this proof, to the point where it now seems almost self-evident, which is certainly not how I've ever felt about this theorem before! Incredible work, subscriber earned.

@vcubingx 5 месяцев назад

Thank you! I'm glad you liked it :)

@cadaeic 6 месяцев назад

Mkt soundtrack with windmills, nothing better

@vcubingx 6 месяцев назад

Nintendo music goes hard

@ingenuity23-yg4ev 6 месяцев назад

oh my god. when you finally joined all the dots and went one by one through each line, it just all clicked and i felt really happy! amazing video!

@justanormalyoutubeuser3868 4 месяца назад

I just discovered your channel and it's absolutely fenomenal. You might have the best combination of rigour and intuition I've seen so far and the wildest thing is how you manage to make these things easy to follow. I'd love to see you cover something from algebraic topology, like number theory it is a subject where theorems are often easy to state but hard to prove (fixed point theorem, Borsuk Ulam theorem, sandwich theorem, hairy ball theorem, four color theorem, Euler's formula...).

@vcubingx 4 месяца назад

Thank you! I've been wanting to cover the four color theorem - I'll get to it eventually :)

@collingallaway5541 6 месяцев назад

While I can’t say I didn’t get lost in a few places, I still enjoyed the video and the concepts! Thanks for breaking the proof down into a comprehensive format

@vcubingx 6 месяцев назад

Thanks for watching!

@charleskolozsvary8714 6 месяцев назад

Gorgeous proof! Will come back to re-digest

@worldhaseverything 6 месяцев назад

Happy to see you again.

@vcubingx 6 месяцев назад

Thank you! Excited to be making videos again :)

@aijzah 6 месяцев назад

omg youre finally back!!

@RohitKavuluru 6 месяцев назад

damn my favorite youtuber is back

@NovaWarrior77 6 месяцев назад

The goat is 🔙

@ElliottKobelansky 6 месяцев назад

I’ve been looking for an explanation of this proof for a while holy ben

@vcubingx 6 месяцев назад

Ben…

@ensiehsafary7633 6 месяцев назад

@@vcubingxHis god is named "Ben"

@forma_de_chapa Месяц назад

Fantastic explanation!

@diribigal 6 месяцев назад

A nice (and shorter, I think) alternative to Mathologer's very similar video!

@vcubingx 6 месяцев назад

Yes! His video is fantastic. To be honest I wouldn't have made this video had I known that it existed...but I was already too deep into making it once I found it :) Hopefully the visuals still added some value!

@martinplesinger 6 месяцев назад

Very well explained. Thanks

@ethanbartiromo2888 6 месяцев назад

The only thing I would recommend is reiterating some points at the end before the conclusion in the sum up. I’m a PhD student in math and while the proof makes sense there were a few questions I had while you were going over it. It was well informed though, so good job!

@randomcubestuff3426 6 месяцев назад

holy ben that last step was so cool 🤯

@vcubingx 6 месяцев назад

rouxles

@morgane3501 6 месяцев назад

loved the proof, really elegant. stopped the video, when you showed it, took me a good 5 mn to fully understand it. i'll watch the video now, but i think i might get this book.

@vcubingx 6 месяцев назад

Thanks!! The book is a must read :)

@ericarcherman9955 6 месяцев назад

Love this proof! Keep it up!

@vcubingx 6 месяцев назад

Thanks eric!

@tunafllsh 6 месяцев назад

The one sentence proof was at no means one sentence though, if you have to break down those non-trivial (to me) steps. Still a beautiful proof.

@jamesyeung3286 6 месяцев назад

We're so back

@hitoshiyamauchi 6 месяцев назад

This is great. Thanks a lot! 😀

@youtubepremium9253 6 месяцев назад

Finally a vid - new subscriber

@susanafaciolince7755 6 месяцев назад

hey, just a tip on accessibility. i noticed by the end of the video and turning up the brightness that you used two different colors in the windmill, but i didn’t notice till then since i have mild colorblindness. that is information that could be loss, so it is a good rule of thumb that, if you want to distinguish colors, they should be very obviously distinguishable by each other.

@PerMortensen 6 месяцев назад

I agree. I could not tell that there two different colors.

@vcubingx 6 месяцев назад

Sorry I didn't address this! I'm honestly a bit ashamed that I didn't consider this when working on this video - I'll work on a better palette for the upcoming videos.

@PerMortensen 6 месяцев назад

@@vcubingx Don't feel too bad, we colorblind people are pretty used to it. It's natural to implicitly assume that others have the same visual experience as oneself. Making sure things are colorblind accessible needs to be a deliberate step in the production process, and you can't take that step if you haven't been made aware of the need.

@Ashen75 6 месяцев назад

Great video! Animations look better.

@vcubingx 6 месяцев назад

Thanks!

@bahachicken89 6 месяцев назад

Great video 👍

@unikracoon1913 6 месяцев назад

What I find so fun is that we spend so long on the cases where z is equal to x when we care about y being equal to x. Throws you in for a loop

@valentinvanhees8690 6 месяцев назад

I have one question; why, in the definition of the windmill set, do we write N^3? (W = {(x,y,z) is in N^3 | x^2+4yz = n}

@primenumberbuster404 6 месяцев назад

Cuz, 4k+1²= 4(k-2)+3² =4(k-6)+5²=4(k-12)+7²=....= 4(k-n²+n)+(2n-1)² =4yz+x²

@thewhitefalcon8539 6 месяцев назад

N means the natural numbers. N^3 means the groups of 3 natural numbers, because N^3 is NxNxN. If you know what a Cartesian product is, that probably seems really obvious now, otherwise it doesn't.

@bjrnstrottman5637 6 месяцев назад

This means that x, y, and z must all be natural numbers. N is the set of natural numbers, so N^3 is the set of ordered triples where all three components are natural numbers. The part after the | gives us another restriction: that the points we are looking for must satisfy the equation x^2 + 4yz = n.

@anibalismaelfermandois6943 6 месяцев назад

Nice Nintendo toons

@LucenProject 6 месяцев назад

I don't know why, but the pokemon music at the start really helped my understanding of Involution!

@vcubingx 6 месяцев назад

Haha, glad you liked it! I especially liked the timing of the song when I was introducing the function and the set

@agesanderas3250 6 месяцев назад

Is that the National Park music? I feel like I haven't played Pokemon in ages.(1:40)

@sagarroy8679 3 месяца назад

Wow! It feels very satisfying to have it all click by the end.

@vcubingx 3 месяца назад

Glad you liked it!!

@cookie_monster171 6 месяцев назад

Hello! I've always had a question bothering me that we had come up with in middle school, but never seemed to find an answer for it although it may be simple. It goes as follows: Let's take a function f(x) (could be real or complex valued) let's start taking the derivatives of f(x), will it jump between functions (like the derivatives of sin(x)) or grow out of control giving a larger and larger formula or even converge to 0? The question is whether there exists an algorithm that tests for that without actually doing the derivatives one by one (so whether it is like Conway's Game of Life or not). I'm sure the answer is pretty trivial but I haven't gotten into maths that much, so I thought I could ask for your and your viewers' help! Thanks in advice!

@sabinrawr 5 месяцев назад

You: proving math Me: jamming to Mario music

@vcubingx 5 месяцев назад

This was me during editing

@alexzhukovsky8361 6 месяцев назад

That was a tough one, especially since I'm not a native English speaker, but after rewinding a bunch of times, it felt amazing finally comprehending this initially gibberish of a proof.

@vcubingx 6 месяцев назад

Glad to hear you understood!

@meofamily4 6 месяцев назад

I have joined that growing population of users of the Internet who watch and listen to math proofs for entertainment. Thank you.

@wesleysuen4140 6 месяцев назад

The proof is so beautiful but only when coupled with this video!

@vcubingx 6 месяцев назад

Thank you!

@beaumatthews6411 6 месяцев назад

What's crazy is I just realized 1/x has this property at work (Math center) when I was trying to find an inverse of it for some reason, then I graphed it on desmos with y=x and was like damn... nice

@roboto12345 6 месяцев назад

This was cool

@antoinedragnir142 6 месяцев назад

This proof is exactly why I love Maths ❤

@astropgn 6 месяцев назад

It is only "one line" if you know the meaning of the words and symbols of that line. For example, what is an involution? So it can only be one line if there is a knowledge agreement between the author of the proof and the reader. This sounds more profound than it really is, since we can all agree that most of our math books had "one line proofs", which were almost always in the form of "the proof is left to the reader". Sounds silly, but if you understand the symbols of the "one liner" in this video, you understand the proof. And if you know the proof of a theorem in a math textbook, then the "the proof is left to the reader" will have the same effect.

@leeashley9772 6 месяцев назад

Happy you are back😊😊

@prdoyle 5 месяцев назад

Ok let me see if I get the gist. The theorem is equivalent to "is there a fixed point on involution A", and to prove there is one, they found a totally different involution B on the same set, proved it has exactly 1 fixed point, and from that, inferred that A must have an odd number of fixed points, ergo it must have at least one?

@miloszforman6270 Месяц назад

Well yes, odd number means it can't be zero.

@TheFakeMackie 9 дней назад

The mario music gave big rolling rock PU vibes.

@miloszforman6270 Месяц назад

This is quite a good presentation of the Zagier proof. The formulation of Zagier looks rather weird. Zagier always takes efforts to present his findings in as terse a way as possible. Which is not good, as far as one is not intending to solve riddles. This "windmill proof" is somewhat original, but I'm not sure if this is really easier to understand than the standard proof, which is an easy consequence of -1 being a quadratic residue (square number) in a prime number field if p has the form 4n+1. Which once again is a consequence of the multiplicative group of p being a cyclic group. The associated math is not that difficult and more straight forward - and it gives more insight into some basics of number theory and algebra.

@beaumatthews6411 6 месяцев назад

8:56... was so confused for a second, I was like 13 is not 3^2+4^2. but i think you meant to say 2^2 (maybe actually just 4)

@Celia_Dawn 6 месяцев назад

I never thought of using Pokemon music to accompany math proofs, but it works wonderfully!

@vcubingx 6 месяцев назад

I love Pokémon music!!!

@maxf9291 6 месяцев назад

Glad I’m not the only one who noticed! Great video

@thewhitefalcon8539 6 месяцев назад

I noticed the Mario music. Copyright strike from Nintendo in 3...2...1...

@Omer-dv2ef 5 месяцев назад

That proof is huge

@faeancestor 6 месяцев назад

I still don't understand how you arrived at 6 million

@vcubingx 6 месяцев назад

Sorry, what do you mean by 6 million?

@thewhitefalcon8539 6 месяцев назад

@@vcubingxthey might be denying the Holocaust. That's my best interpretation...

@krasimirronkov17 6 месяцев назад

Okay but from where does the whole idea of the windmill come from ?

@thewhitefalcon8539 6 месяцев назад

A really clever person thought of it one day, I suppose. This video is showing how the proof proves the result, not how the guy came up with it.

@krasimirronkov17 6 месяцев назад

@@thewhitefalcon8539 ones you have the idea the proof comes natural. So this is a thing related to the video

@swenji9113 6 месяцев назад

That's a very good question. I am by no means a number theorist but I can tell you that it is for sure much more general than this single proof. The polynomials x²+y² and x²+4yz are what we call integral quadratic forms, and it turns out the study of the problem "which integers are sums of 2 squares?" by Fermat played a huge role in identifying the links between number theory and the study of integral quadratic forms. It makes me think about the conic sections: a set of very distinct curves on a plane sharing some properties, that can be way better understood as all possible intersections of a 3-dimensionnal cone with a plane. I guess there's a very similar phenomenon here, where the set of sums of 2 squares can be better described as the intersection of a nice, easier to manipulate, algebraic object in 3 dimensions (the set of windmills) with the plane y=z. My knowledge on the subject is limited but I can recommend this wikipedia page for an insight on the use of quadratic forms in algebraic number theory and its history: en.m.wikipedia.org/wiki/Binary_quadratic_form

@vorrade 5 месяцев назад

"This tells us that 13 is 3 squared plus 4 squared” - minor slip of the tongue there. 8:54 Otherwise a very beautiful video - well worth watching!

@gurmeet0108 6 месяцев назад

@JosBergervoet 6 месяцев назад

This was already posted 3 years ago by Mathologer: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-DjI1NICfjOk.html Some people might find his version clearer (and perhaps some may not...)

@MadDoctorWalsh 5 месяцев назад

This proof is a proper banana sandwich. A complete caboodle. Master work here.

@STEAMerBear 6 месяцев назад

8:58 13 is not 3^2+4^2, it is 3^2+4y^2 for y=z=1.

@krystofsedlacek195 6 месяцев назад

What a proof wow...

@user-fj9hf4bu9f 6 месяцев назад

overall a very informative video thank-you! that being said, could you please reconsider either removing the background music or at the very least making it something less distracting. 👍

@cosimobaldi03 6 месяцев назад

The fact that the Zagier Map is an involution could've been explained a bit better...

@gregbard 6 месяцев назад

This is the only mathematical truth in this sentence. QED.

@michaelgonzalez9058 13 дней назад

Because 4 =1

@wynautvideos4263 6 месяцев назад

You need to develop your own style distinct from 3b1b

@xxx_mouiz_xxx___7969 6 месяцев назад

consider raising the pace a bit

@AttyPatty3 6 месяцев назад

ok i will be honest, i lost you at the second last step of putting it all together " since |Wp| is odd then flip map has odd number of fixed points" I just don't understand..

@vcubingx 6 месяцев назад

When I introduced involutions, I mentioned that the parity of the set is equal to the parity of the number of fixed points! As such, if the size of the set is odd, then any involution as an odd number of fixed points.

@AttyPatty3 6 месяцев назад

@@vcubingx after thinking about it for a bit I think I finally understand, the reason I got confused was I thought of fixed points as if they were just something we keep the same even after the transformation, however know I realized that what you meant by a fixed point was that a point such that f(x) =x. thanks for the proof, it's very satisfying once you get it

@Manigo1743 6 месяцев назад

Drop the music.

@matbru9193 6 месяцев назад

lol one of the authors is the president of my university

@tommasobanfi8133 6 месяцев назад

why is it called "fermat theorem" if it was first proven by euler?

@vcubingx 6 месяцев назад

From Wikipedia: Albert Girard was the first to make the observation, characterizing the positive integers (not necessarily primes) that are expressible as the sum of two squares of positive integers; this was published in 1625.[2][3] The statement that every prime p of the form 4n+1 is the sum of two squares is sometimes called Girard's theorem.[4] For his part, Fermat wrote an elaborate version of the statement (in which he also gave the number of possible expressions of the powers of p as a sum of two squares) in a letter to Marin Mersenne dated December 25, 1640: for this reason this version of the theorem is sometimes called Fermat's Christmas theorem.

@user-nj1og6yb7v 5 месяцев назад

Where (x+y+z)^n=x^2+4yz. Where n=(ln(x^2+4yz))/(ln(x+y+z)). Where (x+y+z)^((ln(x^2+4yz))/(ln(x+y+z))) and where x, y & z are integers.

@mistafizz5195 6 месяцев назад

Bro you're gonna get sued by Nintendo for that OST usage might want to change it before that happens

@vcubingx 6 месяцев назад

Hope not 😭

@vcubingx 6 месяцев назад

I think it should be okay since I used older games which have been OKed by Nintendo…I’ve also used some of these tracks in past videos and it’s been okay

@mistafizz5195 6 месяцев назад

@@vcubingx⁠⁠⁠⁠⁠"I think it should be okay since I used older games which have been OKed by Nintendo..." Famous last words.