Could this be the foundation of Number Theory? The Euclidean Algorithm visualized

[@numbersense3256]

Important corrections at 1:47 Technically Wiles proved something much stronger than what I said in the video. He solved the Diophantine equations x^n + y^n = z^n for all integers of 3 or greater. The case x^3 + y^3 = z^3 (and many other special cases) had actually been solved earlier than this, but up until 1994 the general case seemed almost impossible, even though it had been conjectured hundreds of years earlier by Fermat. But I hope the original point still stands... some Diophantine equations are insanely hard to solve! I also claim something like "Euclid fully solved linear Diophantine equations." This is actually not true. Even though Euclid did describe the Euclidean algorithm, he had not (as far as anyone knows) considered using it to solve linear Diophantine equations in the way I present it in the video. He almost certainly would have been able to if he ever considered it, but it just wasn't the type of question that he was concerned about, since he worked geometrically. On the other hand, Euclid definitely did care about the Euclidean algorithm because it generates the GCD of two numbers. The reference for this is Book VII Prop 1 and 2 of Elements.

[@steviebudden3397]

@Wassim Touyar Yes, but he managed to correct it. Hence it was proved a year later in 1994, as stated.

[@jmckaskle]

Wiles didn't solve the equation xⁿ + yⁿ = zⁿ for n>2, but proved that there are no integer solutions for x, y, and z when n is an integer greater than 2.

[@numbersense3256]

​@@jmckaskle You are correct. Although I meant "solving" in the sense of "finding a proof that characterizes all of the solutions." Sorry if that wasn't clear. Also the equation does have some trivial integer solutions but not if all of the integers are non-zero.

[@x0cx102]

@@jmckaskle that's really a pedantic point. Solving an equation just means finding all solutions and proving that those are the only solutions. If there are no solutions, showing that there are no solutions solves the equation.

[@jmckaskle]

@Roy Long Showing that an equation has no solutions is the opposite of solving an equation. An equation can't be solved if it has no solutions. By saying that Wiles solved xⁿ + yⁿ = zⁿ, for integersx, y, and z and n>2, that gives the impression that there are integer solutions, rather than proving that no integer solutions exist.

[@riddhimanna8437]

So glad to see manim is being used by you and other people now to make wonderful math explainers! Very clear and interesting content! Keep it up!

[@mandeath2971]

Love this video, been self-studying math as a hobby from late 2023, without any tutoring, this is my savior.

[@Grizzly01]

8:50 The Hasse lattice section needs more work, I think, especially some animation to make the concept of 'multiplying in the direction of' clearer.

[@JSIGm]

replacing the plain lines with arrows would make it easier to read

[@jillianonthehudson1739]

New subscriber, looking forwards to more content here! Regardless of how many weeds may or may not be growing in your garden, this was a very nice, clean discussion.

[@eammonful]

Great video! I'll definitely use it if I ever teach discrete math or number theory again. Are you planning on making any more?

[@numbersense3256]

Thanks! I'm hoping to make more if I have the time. I will be posting animations to Twitter too.

[@zpie0333]

This explanation of the Euclidean algorithm is starting to make me truly understands stuff like why the GCD is a linear combination, or how to solve linear congruences

[@frankreashore]

I admit I struggled with this algorithm so it is nice to see it explained visually. Very helpful.

[@ChaoticNeutral6]

Great explanation. Bonus points for including fully worked out examples and taking the time to show how you would generalise them

[@paulensor9984]

Well paced, fun to watch, bravo 👏

[@sethgilbertson2474]

Dude, this is awesome. I'm a 4th grade teacher and can see playing around with these ideas with my higher math kids. More uploads! Can't wait!

[@agustincabrera414]

Great video, was an excelent explanation

[@azurebrown3756]

Euclid was amazing, his Philosophy of Number plays a key role in his writings and theories which still holds as one the foundations of Modern Mathematics as a whole!

[@jaytravis2487]

This video needs more audience members.

[@azurebrown3756]

This is great! Thank you! you should have participated in the 3Blue1bown Visual Math competition!

[@numbersense3256]

This was my submission to the competition!

[@azurebrown3756]

@@numbersense3256 Damn, i wish you would have won

[@sukhjinderkumar2723]

Great stuff!!

[@b.clarenc9517]

8:10 "Any number can be written as a product of primes in exactly one way". I'm not sure I understood this. Did you mean a sum of primes?

[@RodrigoSantos-up8cf]

Nope, the multiplication of primes: 10 = 2 x 5

[@ZweiZombies]

1. GCD (x,y) := largest n such that x/n and y/n are integers 2. if m and n are both solutions, m ≥ n since m is largest and n≥m since n is largest, so n = m remains only solution 3. Euclidean algorithm: a_0= y, a_1 = rem(x,y), a_{i+1} = rem(a_i;a_{i-1}) ...... Begins to scratch head ...... okay a_i exists such that x/a_i and y/a_i are integers .... or does it? Vsauce music starts playing

[@dpatulea]

Wonderful explanation!

[@BenGeorge77]

Yes! Weissman's book is fantastic.

[@ubermensch-mne]

This is great video sir. Please make more.

[@Qermaq]

8:06 I was just going to mention that 2 and 4 will not reach 1, they can only reach their gcd of 2. Now it looks as though that's coming!

[@gnocchi251]

great work

[@thexyouman]

Bravo. More please

[@KakoriGames]

Hold on, at 7:30 you claim that whenever you have a list of numbers that decreases and is non-negative you'll always reach 0, but I think it's important to emphasize that this is only true for integers. If you allow for any real number (which is not the case for the problem in the video), calculus quickly shows us that it is indeed possible for a sequence to get smaller and smaller without ever approaching zero, but have it's limit at a non-zero positive number like 1, 2 or 3/2. Either way, you just got yourself a new sub, hope to see more of you in the future.

[@kwgm8578]

Kakori - you're correct of course, but missing the point. This video is an example of Number Theory, which is the study of integers and integer solutions.

[@Johnarchiebald]

Great video🎉

[@jordanrutledge7943]

Yooo Martie weissman taught my abstract algebra undergrad class. Great teacher, but unfortunately it was the first quarter of covid lockdown so he was forced to teach it online on short notice. I think he did a great job but it would’ve been nice to have some classroom time from him.

[@erawanpencil]

Videos like this make me really wonder if Number Theory, despite seeming so abstract and 'man-made,' may actually eventually lead us to realize that counting, or comparing numbers, is actually something *physical* and objective, not purely mental or subjective... perhaps extremely tiny fluctuations in the EM field or something. There's so much 'structure' in even the simplest arrangement of counts, and our minds are apparently made of electromagnetism after all. I wonder if primitive hominids engraving tally marks on ancient bones in some way intuited this.

[@bernhardbauer5301]

Was it realy Andrew Wiles? Or was it Leonhard Euler 300 years before A. Wiles?

[@abj136]

Correct, Euler solved the case for 3. Wiles solved the general case for n.

[@travisleith1146]

Have I just stumbled upon the beginnings of the next big math channel? Hope so...

[@universallanguageproject]

It's great for a basis and understanding of integers. Great video 👍

[@homoergausster]

i hope you come back someday

[@anilkumarverma240]

GCD is unique:- Assuming c and d are the GCD of (a,b) then a/c=a/d. also b/c=b/d. I.e. a/c=a/d. I.e. c=d

[@AnitaSV]

1:59 it was shown by Euler mostly but first correct one was Kausler (1802), then by Legendre (1820s) etc. Wiles proved for all n that we hadn’t already solved by the

[@benjaminb2934]

great video ! Keep it up !

[@erniesulovic4734]

In Neuro-Linguistic Programming (NLP) which describes how humans operate, we are taught that the only things that are worthy of knowing are context, process and structure. This applies, I have found, to everything in life, including mathematics. Mathematics is designed with context, process and structure......it becomes very simple after that.

[@anilkumarverma240]

The GCD of 2 +be integer is last non zero remainder of a,b such that for every ‘ai’ and ‘bi’ a=b.quotient + remainder. For e.g. GCD (42,30) is as Iteration 1 a=42; b=30; a%b= 12. Iteration 2. a=30; b=12; a%b=6. Iteration 3 a=12; b=6; a%b=0. Since here a%b is 0 Therefore the remainder in iteration 2 is the GCD of (42,30) I.e 6. .... (by definition)

[@sadbinmohshin7564]

If you don't mind can you plz share the book pdf ?

[@ericpham3751]

If one move in space time move against it and one move in time then space move against it too. Just like one expand toward infinity the negative infinity move backward so like breathing lung infinity is the same with small variation

[@shivamkushwahteachingvideo6161]

Nice work

[@judgeomega]

is there a word for a an alternative numbering scheme where numbers are written as a list of its factors and only the primes are given symbols?

[@ramansb8924]

Nice video 👍

[@matthewboyd8689]

This reminds me of the golden ratio If you have a golden ratio number of dots in a circle and you you skip 1/golden ratio of that number you can get to any dot. Neat

[@swartzsteinswartzstein8809]

YES, how did you know this?

[@diophantine1598]

Rather than the hasse diagrams, instead showing how prime factors cancel to reach GCD would be helpful. That’s how I learned to solve GCD anyways.

[@elishmuel1976]

YEah, you got my sub. Super interesting!

[@sayantanbhowmik4196]

now so easy to understand

[@yashagrahari]

Today I also learnt a little bit English with maths.

[@nickallbritton3796]

Thanks for giving homework. Be back soon....

[@brendawilliams8062]

The best. Thx

[@peterboneg]

Euler had already solved n=3.

[@realcygnus]

👍

[@Jkauppa]

what are you foundational operators, and why should you care, only one is good

[@Jkauppa]

many is evil, according to bible

[@Jkauppa]

solve the any number of integers to sum, and the 0 or 1 integer only

[@Jkauppa]

so you have the minimal stepping from the gcd/lcd (a,b), like 2*3*5 = 2*3*7 => 5/7

[@Jkauppa]

I conjure that the prime is only a (0,1) constrained solution set of a general integer (all, not only 0 and 1 multiplier per coefficient) solution set, same for all subsets (0,1) of any number ranges in the knapsack/target problem

[@swartzsteinswartzstein8809]

can you elaborate on what you mean? i do not understand

[@sudip39]

Aryabhatta didn't invented 0 just so you can become one yourself .

[@sayantanbhowmik4196]

improvement needed in the way of explanation

[@ludfde]

GradeAunderA vibes

[@abj136]

I don't find your challenge questions interesting as these were all covered in math class. The video was good though.

[@antoniussugianto7973]

Are you chewing mint gum?

[@frogandspanner]

0:30 Why do many American mathematical KZbinrs, say such things as "negative 6x"? -6x may or may not belong to the set of odd numbers. -6x may or may not belong to the set of even numbers. You do not say "negative even six" - why not? So, why use a set-descriptive adjective ("minus") when referring to a number, but no other set-descriptive adjectives such as "even"? You refer (1:43) to Andrew Wiles, and he refers (kzbin.info/www/bejne/q4LKlKOwgKZooM0) to "plus of minus five", not "positive or negative five". Nowhere does he use "negative" as an adjective for a number, but does use "minus". Is there a good reason that American usage differs from the usage of the person who proved Fermat's Last Theorem?

[@kantaprasadsinha8025]

Europe's hegemony . Diaphanous was great , no doubt. He never told integer solution in linear programma. He made rational solution. First positive integer generalised theory was given by Arya bhatta 1 . See verses 32 and 33, Ganita pada of Aryabhatiya. Except , one or two poems ( ser D E Smith ) on Euclid, Europe vehemently opposed infinity and zero, did not allow number system.. Decarte and Leibnitz were under constant threat for considering 0. Bruno was burnt alive. on 1600 AD. There was no mention of Euclidean algorithms in any book or paper of Algebra before 1950. Now , everything is called Euclidean algorithm.. What name, you suggest for LC M, then you come for name of addition, su traction echt. You find out how Christians hooligans murdered Hypatia ( Theon' s daughter), in 415 AD. My reaction is for your saying diaphantine integer and Euclidean algorithm.

[@Короткоіясно-ь8р]

It's a most obfuscated way ever to explain the Euclidean Algorithm.