Prove that for every prime number $|\{1\leq x \leq p^2 : p^2 | (x^{p-1}-1) \}|=p-1$
26478
Show that the congruence $x^{p-1} \equiv 1 \pmod {p^2}$ has precisely $p-1$ solutions modulo $p^2$
22742
Prove that if $p$ and $p^2+2$ are prime then $p^3+2$ is prime too
121890
Proof that $ p $ | ${ p^n \choose k } $ for any prime $p$ and $ k < p^n$
116387
How to prove that $\Gamma(x) \leq x^x$ for $x \geq 1$
46292
Find all primes $p>2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$
20881
Prove that if $p$ and $q$ are distinct primes such that $pq\mid n^2$ then $pq\mid n$.
54286
Let $p$ a prime number, then prove that : $\sum \limits_{k=0}^{p} \binom {p}{k} \binom{p+k}{k} \equiv 2^p +1 \pmod{p^2}$
91617
Prove this inequality $a \sqrt{1-b^2}+b\sqrt{1-a^2}\le1 $
107774
Prove $\|x\|_1\le \sqrt n \|x\|_2$
36943
How to prove that $L=${$a^p$: p is prime} isn't regular?
88656
For $p$ prime, show that $\Phi_{p^n}(x) = 1 + x^{p^{n-1}} + x^{2p^{n-2}} + \dots +x^{(p-1)^{p^n-1}}$
44019
Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
131385
How to provide a counterexample for if $n$ is prime, then $2^n -1$ is prime
27170
$a\in \mathbb{N}$, $p$ prime, $a<p$ prove that $a\mid p+1\iff\exists\, b,c\in\Bbb N:\dfrac{a}{p}=\dfrac{1}{b}+\dfrac{1}{c}$
64096
Prove that $||z_1|-|z_2|| \leq |z_1-z_2|$.
84603
Prove that if $p$ is a prime such that $p^2+2$ is a prime then $p=3$.
94917
Prove that for every prime $p$, there exists a solution to $(x^2-2)(x^2-6)(x^2-3) \equiv 0 \mod(p)$