This Theorem Has a One-Sentence Proof (Fermat's Christmas/Two-Squares Theorem)

[@Plaquepsoriasis]

I sense a rising star in math KZbin

[@leonardoxnauta]

No way you only have 20 subs man, one of the most well produced math videos ive seen in a while

[@Fire_Axus]

not again…

[@lyrimetacurl0]

Looks like it's taking off now anyway

[@triniasta]

250 now,,,

[@jajajaxdxdxd123]

365

[@teodoricooliveirajunior4140]

402

[@macaroni_slaparoni]

I love this. Your explanations are concise and intuitive. I appreciate that you mentioned how seemingly different areas of math can have hidden connections, it's such a fascinating phenomenon.

[@diribigal]

I agree this was a little fast, and that the equals sign having serifs looks weird, but honestly I'd say it's more important to clean up the sound quality a bit (e.g. with audacity or a new mic or sth). As mentioned in another comment, there is already a Mathologer video that this *particular* video is competing with, but it's still clear you have a passion and skill in presenting mathematics so that I subscribed instantly and am excited to see your next video.

[@Macieks300]

Mathologer also made a very approachable video about this proof of the theorem. I recommend it.

[@ch.6412]

Damn bro your video is great. You're so underrated. Keep it up, we'll wait to see your next video. Very good explanation.

[@isaacgarcesherrero7218]

was studying for an analysis exam and took break to watch your video, keep it up!

[@Rando2101]

I took a look at the one line proof, and it definitely took me some time to get the idea 😅

[@ThuocGiamDau]

Very good visualization !

[@manasthakur1512]

Nice Explanation! I was looking for proof of this.

[@karamboubou8579]

incredible proof and incredible video.

[@rampj4559]

Nice video, keep up the good work!

[@Iovemath]

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 ( kzbin.info/www/bejne/hJKvkHaYaZeKr7s ) 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!

[@Iovemath]

11:16 Agree that math is just soooooo beautiful that I shall spend a lifetime learning and appreciating math!

[@Iovemath]

Looking forward to your new videos!

[@Iovemath]

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?

[@itay_zimet]

Great video, like others said I would maybe slow down the pace a bit and explain the equations more with more dynamic visuals

[@anjalikavyatiwari]

Hi, Loved your video, Btw, I am your 200th subscriber

[@tonibrown148]

I understood the proof the moment I noticed two of the windmills have the same shape. Sometimes having a good visualization changes a lot.

[@leonhan5229]

"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...?"

[@kirbs0001]

This is incredible! If ever this needs a re-upload, or a re-edit, I'd suggest adding little coloured labels to your x,y,z values; labelling them as x,y,z. Could even flash them any time you say 'if y is equal to z' or 'then y equals z' etc.

[@KeNsHoRt47]

bro's thumbnail almost become rejected from art school

[@lukewaite9144]

Please do make more videos this was great :)

[@ishanrana-c9o]

I can already tell this guy will be the next 3blue 1 brown . I lived this video

[@triniasta]

love this guy

[@JayFriendGames3]

Best animation based on math and geometric proof

[@xminty77]

yooo I studied this before, so awesome to see a video for it!!!

[@David280GG]

>sees thumbnail *sigh i can smell the comments

[@draingangarchiveeleven]

thank you math kitty for this blessed video

[@ppppppppppppppppppppppp7]

great video!

[@EcoAging]

Holy goated math video!!!

[@alguem24]

The surprise I got when I saw the sub count Great vid

[@HardikKansara-lt7jd]

My Respect

[@diniaadil6154]

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

[@deleted-something]

Ya’ll love those dark background

[@paridhaxholli]

The thumbnail 💀🇩🇪

[@JJ-fr2ki]

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.

[@anusthanbhandari2766]

Calling it this guy is gonna viral in a month or somethin

[@skmgeek]

great video :3

[@lunars1]

awesome video

[@omerelhagahmed551]

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!

[@poukisgameplays2277]

I was here before 1,000 subs 🥳

[@I_like_smashburgers]

100th sub, congrats!

[@TheEnigmaDreamer]

Brilliant

[@JJ-fr2ki]

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?

[@Zscore6174]

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 (kzbin.info/www/bejne/hJKvkHaYaZeKr7s) since it uses Fermat's Christmas Theorem in the context of Gaussian integers.

[@JJ-fr2ki]

@ wow! Thanks very much.

[@mke344]

New sub!!

[@marcinkoaszewski721]

Can you get a similar result in 3d?

[@Zscore6174]

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.

[@j.dmitrigallow1335]

Really nice video. Love this. Small suggestion: you might consider changing the typeface you use.

[@Zscore6174]

I love the feedback! Do you have suggestions for alternative typefaces?

[@mattgsm]

Didn't Mathologer do a similar video?

[@tox1678]

Its hard to follow, I would concider slowing down the pace to let the audience take the information in.

[@Zscore6174]

Thanks for the feedback! I'll do that for future videos.

[@leif1075]

​​@@Zscore6174thanks but don't you agree it's totally contrived and a cheat tonrewrite B as y times z since no one willl think of that unless they lnew the answer beforehand?? There's just no reason tondp that at all no matter how smart you are. Sonwhy nkt keep it as a squared plus b swuared and since you know one term has to be even jjst rewrite the even term B awiared as 2m squared since you know it's a multiple of two..because.gain there's no reason to rewrite thst one variable as two unless you lnew the outcome.

[@lexiu6036]

eleven minutes is quite a lot for one sentence

[@gratias-tibi-ago]

good video. "most theorems" in description line 1.

[@Zscore6174]

thank you. fixed.

[@ranymattar185]

Very nice and chill video, a very well pronounced explanation and a soothing vibe. you deserve more likes and subs, keep going man!

[@Its_Zer0here]

nice, interesting video, your voice is so calming...☺️ also i like your cat character 😄

[@SevKev-d4z]

Less than 1k subs.... for now. This is incredible man. Dont stop making content u rock

[@jimiwills]

Thats cool ❤

[@tusharchoudhary7880]

This is very intuitive and fun to know about conversion of a mathematical statement into a nice windmill

[@gregbard]

This Theorem Has a One-Sentence Proof. Q.E.D.

[@ricksanchez280]

idk man that was a lot of sentences. Great video tho!

[@Potato2017]

very very very nice video, the only nitpick i have is that i cannot stand this font lol

[@Zscore6174]

Thanks for the input! Do you have suggestions for alternative fonts?

[@aneek1201]

​@@Zscore6174 use a bit more of a round-ish fond. Or maybe just normal text fond like 3blue1brown

[@leif1075]

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?

[@paulbloemen7256]

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.

[@JeguePerneta]

Ok, my engineering brain understood like 20% of what just happened

[@mathladprotocat]

:3

[@sigggo]

You're gonna be famous if you keep this up🫡