Show that $\Phi_{p^n}(X)=\Phi_p(X^{p^{n-1}})$ with $p$ prime.
88656
For $p$ prime, show that $\Phi_{p^n}(x) = 1 + x^{p^{n-1}} + x^{2p^{n-2}} + \dots +x^{(p-1)^{p^n-1}}$
52951
Find all $n \in \mathbb{Z^+}$ such that $\phi(n)=4$
21250
Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$
26233
$[K(\zeta_n):K]=\phi(n)$?
158106
Prove that 1 less than the number of equivalence classes divides $p-1$ where $p$ is prime
34491
Is there a way to show that $\sqrt{p_{n}} < n$?
95111
Elementary proof for $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ where $p_i$ are different prime numbers.
30630
Proving ${n \choose p} \equiv \Bigl[\frac{n}{p}\Bigr] \ (\text{mod} \ p)$
121723
$\phi(n)=\frac{n}{2}$ if and only if $n=2^k$ for some positive integer k
71788
Prove existence of multiplicative inverse modulo p when p is prime
58182
How to prove that $\mathbb{Q}[\sqrt{p_1}, \sqrt{p_2}, \ldots,\sqrt{p_n} ] = \mathbb{Q}[\sqrt{p_1}+ \sqrt{p_2}+\cdots + \sqrt{p_n}]$, for $p_i$ prime?
37346
If $m, n$ be positive integers, prove that $\phi(mn)=\phi((m,n))\phi([m,n])$, where $(m,n)=$ gcd of $m, n$ and $[m, n]=$ lcm of $m, n$.
83752
If $p + q = 1$ prove that for any natural $n, m$ following is true: $(1 - p^n)^m + (1 - q^m)^n \ge 1$
103360
Prove $4p-3$ is a square knowing that $n\mid p-1$ and $p\mid n^3-1$, $p$ prime
27170
$a\in \mathbb{N}$, $p$ prime, $a<p$ prove that $a\mid p+1\iff\exists\, b,c\in\Bbb N:\dfrac{a}{p}=\dfrac{1}{b}+\dfrac{1}{c}$
152299
Proving $a^{n-1} \equiv 1 \mod n $ when n is not prime.
149509
What method would you go through to find an $n$ such that $\phi(n)=100$?