$p$ and $q$ are primes. Prove $\forall n,k\in \mathbb N, (p^n\mid q^k⇒p=q)$
68221
Question regardles primes and the fundamental theorem of arithmetic
66851
If $p,q$ are primes $(pq,a)=1$ and $p-1|q-1$ then prove $a^{q-1}\equiv 1\pmod{pq}$
30374
If $n\in\mathbb N$ and $4^n+2^n+1$ is prime, prove that there exists an $m\in\mathbb N\cup\{0\}$ such that $n=3^m$.
10403
Let $p$ be a prime integer. Show that for each $a ∈ GF(p)$ there exist elements $b$ and $c$ of $GF(p)$ satisfying $a = b^2 + c^2$.
131488
Show that $p^3+4$ is prime
106401
Prove: $p-mq | f(m)$ where $m$ is any integer
103496
Show that $3p^2=q^2$ implies $3|p$ and $3|q$
61103
Prove that $\mathbb{|Q| = |Q\times Q|}$
110876
Prove that $p\mid \binom{p}{k},\ 0< k< p$
34692
Prove if $ p$ is a prime of the form $3n+1$, then $p$ is also of the form $6m+1$
70066
Show that product of primes, $\prod_{k=1}^{\pi(n)} p_k < 4^n$
17601
How to prove that if $a\equiv b \pmod{kn}$ then $a^k\equiv b^k \pmod{k^2n}$
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?
64782
How to prove if $n$ is prime and $n | a^2$ then $n | a$?
140294
prove that $n >2$ and $n$ is a prime then $n$ is odd
39843
Let $P$ be a prime. Show that $\exists$ $x \in \mathbb{N}$ such that $f(x) = p$ then $\exists$ $y \in \mathbb{N}$ such that $g(y) = p$
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.