Let $\phi$ denote Euler's totient function, $p > 1$ be prime, and $n\in\N$. Conjecture and prove a formula for $\phi(p^n)$ in terms of $p$ and $n$.

Let $\phi$ denote Euler's totient function, and $p, q > 1$ be primes. Conjecture and prove a formula for $\phi(pq)$ in terms of $p$ and $q$.

In the context of RSA, what is the significance of Euler's Totient Function (φ(n))?It determines the set of co-primes less than n.It is used to compute the public key.It is crucial for calculating the private key.It defines the block size for encryption.

\begin{problem} For $n\in\N$, let $\sigma_k(n)$ be defined as the sum of the $k$th power of the positive divisors of $n$, i.e.: \[ \sigma_k(n) = \sum\limits_{d|n} d^k. \] Let $p$ be prime and $e\in \N_0$. Through experimentation, develop conjectures for the following: \begin{enumerate} \item $\sigma_0(p^e)$ \item $\sigma_1(p)$ \item $\sigma_1(p^e)$ \end{enumerate} Then, choose one of the above to prove.\end{problem}\begin{proof}\end{proof}

If n is prime then for every x 0<x<n, (xn-1 % n)=1 according to?infoYou have max 2 attempts to score in this question.Attempts left:2/2OptionsThis problem has only one correct answerFermat’s theoremWilson’s theoremMiller rabinNone of the above

There is an integer n > 5 such that 2^n − 1 is prime. Can you prove this?