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!
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
If you're referring to the statement "xy is coprime to n if and only if x and y are both coprime to n", then yes, that is true. This is a consequence of Euclid's lemma, which states that if a prime p divides xy, then p must also divide at least one of x or y. Conversely, if p divides x or y, then clearly it also divides xy, since xy contains the prime factors of both x and y.
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?
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.
= 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. !!!!!!!!!!!!@@@@@@@!!!!!!!!!!!!!!!!!
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.
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.
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
![Lp. Сердце Вселенной #60 РОЖДЕНИЕ ЛОЛОЛОШКИ [Финал] • Майнкрафт](http://i.ytimg.com/vi/YoR0pAV9FVQ/mqdefault.jpg)
If the notation at 9:07 is confusing, I talk a little about what it means in this video: ua-cam.com/video/SslPWR2N5jA/v-deo.html
precise explanation of totient theorem I've ever come across
Best, most clear, concise least stuttery explanation I've ever seen of the Proofs of the Totient Theorem. Also amazing proof of FLT. Congrats!!
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!
Great Video. The concept seemed so abstract without seeing the intuition behind it. Now it makes total sense!
The best demonstration I ever saw about this ! Thank you
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
So straightforward! Good Lecturer
Thank you! Is the best proof of this theorem I've seen!
nice video i was looking for a neat proof of the totient formula for a while now and you did it!
bro really made me think fr, amazing video
Thank you for the delicate demonstration
Amazing video. Clear and detailed explanations.
your really good at speaking
Fantastic proof
I can understand this with secondary school math, meaning a great job done by you. Thanks
I'm merely struggling with this and can't even understand a thing. Thank you sooo much for keeping everything straight
Love you Sir. You are helping people
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?
The elements of S are pairwise incongruent to each other mod n because (by definition) they are distinct numbers between 1 and n.
Only one word "Awesome"💌
Very clear explanation! Great!!!
Very neat. Amazing work. Thank you!
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
2:30 is there an equivalence? or is it just an implication?
If you're referring to the statement "xy is coprime to n if and only if x and y are both coprime to n", then yes, that is true. This is a consequence of Euclid's lemma, which states that if a prime p divides xy, then p must also divide at least one of x or y. Conversely, if p divides x or y, then clearly it also divides xy, since xy contains the prime factors of both x and y.
Thank you! A very good proof
You're great!! Keep making more videos :D
thank you so much, great and very helpful video
Excellent video! thank you
Excuse me how do you know that all element in T’ are different from other?
(Ex ax2 = ax3 (mod n) some thing else like that)
See 3:32-4:56.
Thank you so much it really helped
The injection part... if we define f(x) = ax mod n, and gcd(a,n) = 1 , then if x1 =/= x2 , f(x1) =/= f(x2)
Thank you so so much
This lecture was so amazing, that it seems like you should be charging a phi!
It took me a minute to remember that phi has two pronunciations! Lol
@@MuPrimeMath When it comes to pronunciation, I just go with what sounds right and don't try to phite it 👍
I could not understand injection mod n
Which college have you choose?
Caltech!
@@MuPrimeMath congratulations and all the best!!
thanks so much ! just a small question : is saying that a isn't multiple of p, the same as gcd(a,p) = 1 ?
Yes, because p is prime.
@@MuPrimeMath thanks so much !
thank you
You sound a bit like Addison Anderson who narrates the tedX videos!
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?
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.
@@MuPrimeMath does that answer to my questions imply your in high school or just graduated? Yes or no?
I'm in my freshman year of college.
@@MuPrimeMath ok🥸
Don't forget to take Putnam Comp or equivalent as undergrad. Best of luck.
= 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. !!!!!!!!!!!!@@@@@@@!!!!!!!!!!!!!!!!!
What if a is a negative integer
The theorem is still true!
@@MuPrimeMath so will the remainders still be from 1 to n-1 because i dont think so
The remainder of any integer when divided by any positive integer is always between 0 and n-1.
@@MuPrimeMath but what about negative remainders
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.
Would I get a higher mark in my meth test if I wrap my head with a towel?
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.
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
bro is so pretty i can not focus on the board
your profile picture missed an S