$n = 2^p - 1$ Prove that if n is prime then p must also be prime.
41700
Are Mersenne prime exponents always odd?
158563
If $m^4+4^n$ is prime, then $m=n=1$ or $m$ is odd and $n$ even
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.
66851
If $p,q$ are primes $(pq,a)=1$ and $p-1|q-1$ then prove $a^{q-1}\equiv 1\pmod{pq}$
143772
How can I algebraically prove that $2^n - 1$ is not always prime?
140412
If $p$ and $q$ are distinct primes such that $p|c$ and $q|c$, prove that $pq|c$
135586
Prove that the preimage of a prime ideal is also prime.
50781
$(n+1)$th prime $p_{n+1}$ less than or equal to $p_1p_2\dots p_n+1$
97021
Do there exist two primes $p<q$ such that $p^n-1\mid q^n-1$ for infinitely many $n$?
158106
Prove that 1 less than the number of equivalence classes divides $p-1$ where $p$ is prime
65826
Find a prime number $p$ and an integer $b<p$ such that $p$ divides $b^{p−1}−1$.
54547
Prove this number must be a prime number or 1.
26503
Suppose that $p$ ≥ $q$ ≥ $5$ are both prime numbers. Prove that 24 divides ($p^2 − q^2$)
84603
Prove that if $p$ is a prime such that $p^2+2$ is a prime then $p=3$.
111037
Consider numbers of the type $n=2^m+1$. Prove that such an n is prime only if $n=F_K$ for some $ k \in N$, where $F_k$ is a Fermat Prime.
68328
Infinitely many $n$ such that $2^n-n$ is divisible by a prime
119051
Show that only finitely many positive integers $n$ have the property that for all $m$ for which $1<m<n$ and $\text{gcd}(m,n) = 1$, $m$ is prime.