Explicit Formula for Euler's Totient Function!

[@KajetanOlas]

The way you explain these concepts is perfect. After watching your videos I understand efortlessly where the theorems come from. When listening to other lectures I'm usually able to follow through as well but I need to concentrate hard all the time and It's tiring. Watching you feels like watching Richard Feynman. Thanks a lot!!!

[@maynardtrendle820]

This channel is great. Just found it during the megafav number event. Keep up the great work!

[@fracaralho]

This demonstration is really cool. One could also arrive at the same point by thinking: if my number n has a prime p among its factors, then no multiple of p can be coprime to n. But, during counting, a multiple of any number k occurs once every k numbers. So you can expect n/k multiples of k from 1 to n. Naturally, this also applies to prime numbers. This means that n - n/p numbers won't share a factor p with n. If you extend this line of reasoning to all prime factors of n, you get exactly the formula for the totient function. For example: 24 has 2 and 3 among its prime factors. So no multiples of 2 or 3 can be coprime to 24. But 1/2 of natural numbers are multiples of 2, since even numbers occur every two consecutive numbers. And 1/3 of natural numbers are multiples of 3, since a multiple of 3 occurs every three consecutive numbers (yeah, I know it isn't rigorous of me to speak of halves and thirds of infinite sets, but, please, bear with me). So one half of the numbers between 1 and 24 will share a factor of 2 with it, and one third of the numbers will share a factor of 3. So one half and two thirds of numbers, respectively, *won't* share a factor of 2 or 3 with 24. So we find that 24 * (1/2) * (2/3) = 8 numbers will be coprime to it. But this is precisely the definition of the totient function!

[@MuPrimeMath]

You do have to pay attention when talking about that kind of stuff! The process that you're describing is similar to an alternate method to prove that φ(ab)=φ(a)φ(b). The important thing to remember is that it's possible for a number to be BOTH a multiple of 2 and a multiple of 3, so we can't immediately assume that we remove 1/2 in the first step and 1/3 in the second step. Ultimately the proof does work out, but you have to prove that after removing all the multiples of 2, a third of the REMAINING numbers are multiples of 3, and so on. It's an interesting exercise to think about how to prove that part!

[@EpicMathTime]

Left-handed crew oh wait.. _(accidentally exposes himself)_

[@diweiye8420]

clearest video i've seen so far 👍

[@Naoseinaosei213]

I would like a vídeo about Gauss theorem.

[@Naoseinaosei213]

Nice vídeo.

[@ezras7997]

Factorization is weird, still very neat, also there were only two splices that I saw, also very nice

[@guill3978]

How do you calculate the value of the euler's toilent function without having to find the factorization of n?

[@MuPrimeMath]

This is probably not possible, since the value of the totient function depends on its prime factors.

[@peachesaupear8455]

@@MuPrimeMath Check out these guys video for #SoME. Astoundingly, they found a formula for the Totient function without needing to know a number's prime factors. kzbin.info/www/bejne/n6TJZ56Vh7uJbK8

[@sanelprtenjaca9776]

Very nice. Gauss theorem next! :)

[@gesm392]

Watching it from Mexico.

[@aliberro]

Best of the best ❤️♥️

[@thayanithirk1784]

Love from india

[@djvalentedochp]

👍👍👍👍👍👍👍👍👍👍

[@shivimish9962]

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