Prove that if $n\ge4$ then $n,n+2,n+4$ cannot all be primes.
71110
Show that we cannot have a prime triplet of the form $p$, $p + 2$, $p + 4$ for $p >3$
75733
Divisibility of prime numbers
123033
Showing a number $n$ is prime if $(n-2)! \equiv 1 \pmod n$
64782
How to prove if $n$ is prime and $n | a^2$ then $n | a$?
75378
how to prove this extended prime number theorem?
102889
If $2^n - 1$ is prime from some integer $n$, prove that n must also be prime.
151701
Does there exist an $a_0$ such that the sequence $a_{n+1} = 2a_n + 1$ is prime for all $n \ge 0$?
119486
Upper and Lower bounds on the nth prime number
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.
44857
Why the $(n+1)$th prime number cannot exceed the smallest prime divisor of $p_1\dotsc p_n + 1$
130846
Prove or disprove: If $a \in \mathbb{Z}$, $a > 0$, and $p$ is a prime such that $p^4 \mid a^3$, then $p^2 \mid a$.
19761
Show for prime numbers of the form $p=4n+1$,  $x=(2n)!$ solves the congruence $x^2\equiv-1 \pmod p$. $p$ is therefore not a gaussian prime.
34954
If $11..11$($n$ times 1) is prime, prove that $n$ is also prime
29572
Proof that $2^n-1$ does not always generate primes when primes are plugged in for $n$?
40565
How can I prove that one of $n$, $n+2$, and $n+4$ must be divisible by three, for any $n\in\mathbb{N}$
131488
Show that $p^3+4$ is prime
38805
Prove that $x^{2} \equiv -1 \pmod p$ has no solutions if prime $p \equiv 3 \pmod 4$.