Questions tagged [totient-function]

Questions on the totient function $\phi(n)$ (sometimes $\varphi(n)$) of Euler, the function that counts the number of positive integers relatively prime to and less than or equal to $n$.

The Euler phi function $\phi(n)$ is defined to be the number of integers between $1$ and $n$ which are relatively prime to $n$; that is,

$$\phi(n) = |\{1 \le k < n : \gcd{(k, n)} = 1\}|$$

The phi function is multiplicative; that is, if $n$ and $m$ are relatively prime, then

$$\phi(nm) = \phi(n) \phi(m)$$

There is also a product representation,

$$\phi(n) = n \prod_{p|n} \left(1 - \frac{1}{p}\right)$$

where the product is taken over prime divisors of $n$.

One particularly important use of Euler's phi function is in computing exponents with modular arithmetic. Whenever $a$ and $n$ are relatively prime, Euler's theorem states that

$$a^{\phi(n)} \equiv 1 \pmod{n}$$

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Why is Euler's Totient Theorem right?

I am trying to understand Euler's Totient Theorem but I don't understand why it works: $$m^{\phi(n)}\equiv1 \text{ mod } n$$ Where m and n are coprime, how can a number m to the power of phi(n) be congruent to 1 mod n. I mean, in the…
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Does there exist a number for which euler phi of $9n$ equals $91$?

Does there exist such $n$ that euler phi for $9n$ is equal $91$?
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What's kind of integer n such that $\varphi(n)>n$?

I want to know which integer $n$ has a property that $\varphi(n)>\dfrac{n}{2}$, where $\varphi(n)$ is the Euler's totient function. Thank you.
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