Euler's Totient Theorem and Fermat's Little Theorem - Complete Proof & Intuition

[@MuPrimeMath]

If the notation at 9:07 is confusing, I talk a little about what it means in this video: kzbin.info/www/bejne/iaTPgYqIZ7NooKM

[@gautamiwankhede6357]

precise explanation of totient theorem I've ever come across

[@MrKrabs-xf2tr]

Best, most clear, concise least stuttery explanation I've ever seen of the Proofs of the Totient Theorem. Also amazing proof of FLT. Congrats!!

[@martinsandor707]

Jesus christ man I started looking into modular arithmetic without knowing anything about it beforehand as part of a high school assignment, and your clear explanations helped me understand everything in such a way, that even 5-10 other videos couldn't accomplish! Keep up the good work, you deserve so many more views and subs!

[@user-ou1yw4cg6y]

Thank you for the delicate demonstration

[@dandervi6196]

Great Video. The concept seemed so abstract without seeing the intuition behind it. Now it makes total sense!

[@gergodenes6360]

This is a pretty neat proof of these theorems! It's simple, straightforward, and not at all tedious, good job! Do you plan on doing a video on the Chinese remainder theorem?

[@MuPrimeMath]

I don't plan to make a video on the Chinese remainder theorem, but it will be important for one of the derivations coming up soon. I will make a community post about it in the next few days!

[@leylin3161]

Where can i check the post?

[@MuPrimeMath]

The community post is mainly a link to this playlist: kzbin.info/aero/PL22w63XsKjqyg3TEfDGsWoMQgWMUMjYhl I recommend watching the first two videos.

[@egillandersson1780]

The best demonstration I ever saw about this ! Thank you

[@christianrudder4826]

That was absolutely insane, my text book really had a field day with making it extremely complicated by using maps. This was clear and I was able to see the magic, I was blown away by the result of this proof

[@vaginalarthritis1753]

12:24 Oh ho ho! I know that excitement in your voice - i've had that excitement in my voice - where everything becomes clear and the pieces fall in place, where you see the light at the end of the tunnel. It's beautiful, Good stuff.

[@stanees9657]

So straightforward! Good Lecturer

[@spartansai1107]

Amazing video. Clear and detailed explanations.

[@xxxprawn8374]

nice video i was looking for a neat proof of the totient formula for a while now and you did it!

[@toni_canada]

Thank you! Is the best proof of this theorem I've seen!

[@Heidi-fz2lg]

I'm merely struggling with this and can't even understand a thing. Thank you sooo much for keeping everything straight

[@felipemurta9160]

bro really made me think fr, amazing video

[@ronnie2367]

I can understand this with secondary school math, meaning a great job done by you. Thanks

[@ttrss]

your really good at speaking

[@dvoorganisation459]

Love you Sir. You are helping people

[@chap_eau]

thank you so much, great and very helpful video

[@eladiomilad8500]

Thank you so so much

[@mathtutor5110]

Very clear explanation! Great!!!

[@iamavolk]

Very neat. Amazing work. Thank you!

[@ritwikpriyadarshi961]

Thank you! A very good proof

[@PunmasterSTP]

This lecture was so amazing, that it seems like you should be charging a phi!

[@MuPrimeMath]

It took me a minute to remember that phi has two pronunciations! Lol

[@PunmasterSTP]

@@MuPrimeMath When it comes to pronunciation, I just go with what sounds right and don't try to phite it 👍

[@sharonsavyon1384]

Excellent video! thank you

[@xoppa09]

The injection part... if we define f(x) = ax mod n, and gcd(a,n) = 1 , then if x1 =/= x2 , f(x1) =/= f(x2)

[@ring5148]

thank you

[@xoppa09]

i think there is a small typo at 11:35. it should be Product (x_i_j ) = Product a * x _ j , since j ranges from 1 to phi

[@vishalarun1288]

4:20 Why is set S not congruent mod n. How does it being between 1 and n make it incongruent? and what is it incongruent to?

[@MuPrimeMath]

The elements of S are pairwise incongruent to each other mod n because (by definition) they are distinct numbers between 1 and n.

[@non_sense7441]

You're great!! Keep making more videos :D

[@sudhanvabharadwaj2765]

Thank you so much it really helped

[@prakharpandey8968]

I could not understand injection mod n

[@kiaruna]

thanks so much ! just a small question : is saying that a isn't multiple of p, the same as gcd(a,p) = 1 ?

[@MuPrimeMath]

Yes, because p is prime.

[@kiaruna]

@@MuPrimeMath thanks so much !

[@EpicVideos2]

You sound a bit like Addison Anderson who narrates the tedX videos!

[@robertflynn6686]

I am curious here as to why you are putting out videos on so many varied math topics? Is this a project? Anyway good luck going to Caltech. If I can ask you, what for you is the most interesting work going on right now. In science or math or other?

[@MuPrimeMath]

I'm not sure what I find most interesting yet because I still don't have much background knowledge! I hope to learn more math so that I can understand what's happening at the frontiers of math research.

[@robertflynn6686]

@@MuPrimeMath does that answer to my questions imply your in high school or just graduated? Yes or no?

[@MuPrimeMath]

I'm in my freshman year of college.

[@robertflynn6686]

@@MuPrimeMath ok🥸 Don't forget to take Putnam Comp or equivalent as undergrad. Best of luck.

[@odinedin8265]

= THE GREAT! - THE GREATEST!!! Theorem of the 21st century! = !!!!!!!!!!!!!!!!!!!!! "- an equation of the form X**m + Y**n = Z**k , where m != n != k - any integer(unequal "!=") numbers greater than 2 , - INSOLVable! in integers". !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! /- open publication priority of 22/07/2022 / /-Proven by me! minimum-less than 7-10 pp. !!!!!!!!!!!!@@@@@@@!!!!!!!!!!!!!!!!!

[@dharmanshah1239]

Which college have you choose?

[@MuPrimeMath]

Caltech!

[@dharmanshah1239]

@@MuPrimeMath congratulations and all the best!!

[@asriel522]

Would I get a higher mark in my meth test if I wrap my head with a towel?

[@fahrenheit2101]

Oh wow, I knew about Fermat's little theorem but I had no idea it was a special case of this one. Cool video, but I might need to get slightly more comfortable with some of the fundamentals of modular arithmetic to completely get the rules, though you seem to have videos for that too, which helps.

[@krasimirronkov17]

What if a is a negative integer

[@MuPrimeMath]

The theorem is still true!

[@krasimirronkov17]

@@MuPrimeMath so will the remainders still be from 1 to n-1 because i dont think so

[@MuPrimeMath]

The remainder of any integer when divided by any positive integer is always between 0 and n-1.

[@krasimirronkov17]

@@MuPrimeMath but what about negative remainders

[@MuPrimeMath]

The standard definition of remainder ( en.m.wikipedia.org/wiki/Euclidean_division#Division_theorem ) defines a remainder to be non-negative. A negative number is always congruent to a positive number mod n for n>0, so a negative remainder is always equivalent to some positive remainder.

[@ireallyhateweekends]

bro is so pretty i can not focus on the board

[@s00s77]

unrelated comment, but it's worrying me is youtube under some bot raid? arthritis' styled comments popping everywhere at every "small" channel i visited and could read through the comment section

[@ynot8319]

your profile picture missed an S