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

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Video on coprime numbers mod n: • Coprime Numbers and Re...
Video on the cancellation rule for modular arithmetic: • When can we cancel a f...
Euler's theorem relates to modular exponentiation. Fermat's little theorem is a special case for prime modulus. Here we go through an explanation of the derivation for Euler's totient theorem so we can understand where it comes from!
Modular Arithmetic playlist: • Modular Arithmetic
0:00 Euler's Totient Theorem
13:35 Fermat's Little Theorem
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Music: OcularNebula - The Lopez

Опубликовано:

30 июл 2024

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Комментарии : 63

@MuPrimeMath 3 года назад

If the notation at 9:07 is confusing, I talk a little about what it means in this video: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-SslPWR2N5jA.html

@gautamiwankhede6357 3 месяца назад

precise explanation of totient theorem I've ever come across

@MrKrabs-xf2tr 3 года назад

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 3 года назад

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!

@egillandersson1780 3 года назад

The best demonstration I ever saw about this ! Thank you

@dandervi6196 2 года назад

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

@xxxprawn8374 3 года назад

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

@vaginalarthritis1753 3 года назад

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.

@spartansai1107 3 года назад

Amazing video. Clear and detailed explanations.

@toni_canada Год назад

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

@user-ou1yw4cg6y Месяц назад

Thank you for the delicate demonstration

@_avr314 2 года назад

Very neat. Amazing work. Thank you!

@stanees9657 Год назад

So straightforward! Good Lecturer

@christianrudder4826 9 месяцев назад

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

@mathtutor5110 3 года назад

Very clear explanation! Great!!!

@ronnie2367 3 года назад

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

@dvoorganisation459 3 года назад

Love you Sir. You are helping people

@non_sense7441 3 года назад

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

@felipemurta9160 Год назад

bro really made me think fr, amazing video

@sharonsavyon1384 3 года назад

Excellent video! thank you

@ritwikpriyadarshi961 2 года назад

Thank you! A very good proof

@gergodenes6360 3 года назад

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 3 года назад

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 8 месяцев назад

Where can i check the post?

@MuPrimeMath 8 месяцев назад

The community post is mainly a link to this playlist: ru-vid.com/group/PL22w63XsKjqyg3TEfDGsWoMQgWMUMjYhl I recommend watching the first two videos.

@chap_eau 6 месяцев назад

thank you so much, great and very helpful video

@sudhanvabharadwaj2765 3 года назад

Thank you so much it really helped

@eladiomilad8500 Год назад

Thank you so so much

@Heidi-fz2lg 2 года назад

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

@ttrss Год назад

your really good at speaking

@ring5148 9 месяцев назад

thank you

@xoppa09 3 года назад

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

@xoppa09 3 года назад

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

@PunmasterSTP 3 месяца назад

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

@MuPrimeMath 3 месяца назад

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

@PunmasterSTP 3 месяца назад

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

@EpicVideos2 3 года назад

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

@vishalarun1288 8 месяцев назад

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 8 месяцев назад

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

@kiaruna Год назад

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

@MuPrimeMath Год назад

Yes, because p is prime.

@kiaruna Год назад

@@MuPrimeMath thanks so much !

@prakharpandey8968 Месяц назад

I could not understand injection mod n

@fahrenheit2101 2 года назад

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.

@dharmanshah1239 3 года назад

Which college have you choose?

@MuPrimeMath 3 года назад

Caltech!

@dharmanshah1239 3 года назад

@@MuPrimeMath congratulations and all the best!!

@robertflynn6686 3 года назад

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 3 года назад

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 3 года назад

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

@MuPrimeMath 3 года назад

I'm in my freshman year of college.

@robertflynn6686 3 года назад

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

@odinedin8265 2 года назад

= 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. !!!!!!!!!!!!@@@@@@@!!!!!!!!!!!!!!!!!

@krasimirronkov17 2 года назад

What if a is a negative integer

@MuPrimeMath 2 года назад

The theorem is still true!

@krasimirronkov17 2 года назад

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

@MuPrimeMath 2 года назад

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

@krasimirronkov17 2 года назад

@@MuPrimeMath but what about negative remainders

@MuPrimeMath 2 года назад

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.

@asriel522 2 года назад

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

@ireallyhateweekends 5 месяцев назад

bro is so pretty i can not focus on the board

@s00s77 3 года назад

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