Introduction to Euler's Totient Function!

Euler's totient function φ(n) is an important function in number theory. Here we go over the basics of the definition of the totient function as well as the value for prime numbers and powers of prime numbers!

0:00 Definition and example
1:56 Totient of a prime number
3:30 Totient of power of a prime

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Music: OcularNebula - The Lopez

[xxxprawn]

I was researching this when I was preparing for olympiads this year :) Very well explained!! Keep it up!

[chrisgreen_]

I love your videos! They're so clear and well-explained.

[aimeerimturatbekova]

It was very helpful, thanks a lot!!!!

[AaronHollander]

Why do we consider n as having a property that it can never have? The first criteria should be to check all numbers up to n-1.

[mohammedno]

Thank you so much for this video it help me :)

[JohnSmith-xvxy]

Great video! Very good explanation. Can you please clarify one example:

Let's say we have 3^6=729. According to the video, we should have 3^5=243 integers (less than or equal to 3^6) that have common divisor/factor with 729 (basically, multiples of 3). Does it mean that the number of co-prime integers is 729-243=486?

[klaybond]

I wished my math teacher had explained it like this

[ZackSussmanMusic]

Why do we count 1 as a non-shared factor even when 5= 5*1 and 6 = 6*1

[CrashExhibition]

I am missing one basic but crucial fact due to dumbness. We use P^(a-1) as the common factor, why is it not P^a? I understand how to factor it out and get the result, but I don’t really understand why we choose it as a common factor.

[MuhammadTaimur]

Shouldn't the prime factors of p^a be p^1, p^2, instead of p, 2p, 3p, 4p...?

[razvanbirisan]

Without Euler's product formula? 😥

[jasonhuang]

It's a little hard to understand but still ok. Thank you!!