Number Theory | Euler's Theorem Proof

We present a proof of Euler's Theorem.

[georgesadler]

Professor Michael Penn, thank you for an outstanding proof of Euler Theorem.

[N]

I like a slightly different version of this proof, where we take the function f_a:S->S which maps x to (ax)mod n, check it's well defined, see that it's inverse is f_{a^-1} and we get that every element in S can be "written" as ax, with a unique x, so we continue and do the final part.

[vishalmishra]

Simple proof. Great video. Thanks Michael. (For some reason, I was expecting a more complex proof, given the structure of this theorem)

[fredpim]

so glad to have again 61 videos to watch

[skleon]

One question: in 6:56 you use the fact that if a*b = 0 (mod n), then n must divide a or n must divide b. This is clearly true if n is prime. But for non primes it can be false. Example: 2 * 4 = 0 (mod 8). So how do we know that in this case, this statement is valid?

[tianlouw]

7:00
Why?
If n=10, a=5, xi=7 and xj=3
Then 10|5(7-3) holds
10 does not divide 5
But according to him, this means 10|(7-3) ie. 10|4 which is false.
n does not have to be prime, 5, 7 and 3 are coprime to n. SO what am I missing?

[tisyarawat]

Could you please make a video on de Polignac's Formula? It would be really nice if you could make one explaining the formula along with a few sums as examples. :)

[guogeorge]

A year after, for the second time I watch this video, already knowing Ferman's Little theorem, I finally get the ins and outs.

[yiliangliang]

Hi, why does it follow from (1) and (2) that S and aS are congruent sets mod n? Thx!

[jungwookrlee]

would this be the same thing as proving that set of euler's numbers, S = {1 ≤ x ≤ n | gcd(x, n = 1} and set the of the residues, R = { 0 ≤ y < n | ax = y mod n} are the same set? (using the definition of set equality?)

[sesar_]

Third times the charm for understanding. Thanks!

[deepankulandaisami]

So clear and understandable. thank you so much!

[awaiskhan]

Correct me if I'm wrong but, but isn't the property n| a(xi -xj) implies n
divides a or n divides xi-xj is only applicable when n is a prime number, but in our proof we are assuming n to be an arbitrary natural number?

[snipergranola]

I love your videos just studying group theory

[anantsingh]

Students- Last couple of lines
Teachers- Last couple of pages
Legends- Last couple of BOARDS
8:30






Thanks for the proof . Very less proofs of such topics are available generally.From india 🙏

[qedmath]

could someone explain in detail why p has to be prime and why p can not divide a?

[AnuragSingh-monb]

i am sad as you didn't said in the end "which is a good place to stop"

[TheStargazer]

You are quite brilliant! I empathize with your angry sinuses but you don’t look like it’s any issue!

[mehrajulabadin]

In 7:11 why Xi and Xj both are smaller than n ?

[guogeorge]

key point is the two sides of the equation share no common factor, so they cancels out