Prove that if $n\geq 4$ is even then $2^n -1$ is not prime.
41700
Are Mersenne prime exponents always odd?
54359
Odd Primes Problem Proof
62982
Prove $2^b-1$ does not divide $2^a + 1$ for $a,b>2$
100558
Prove that for any even positive integer $n$, $n^2-1 \mid 2^{n!}-1$
134963
Proving a non-prime (composite) number $> 1$ has a divisor $d \leq\sqrt q$
158563
If $m^4+4^n$ is prime, then $m=n=1$ or $m$ is odd and $n$ even
154475
Prove that if $2^n - 1$ is prime, then $n$ is prime for $n$ being a natural number
38805
Prove that $x^{2} \equiv -1 \pmod p$ has no solutions if prime $p \equiv 3 \pmod 4$.
80658
Prove that the number $19\cdot8^n+17$ is not prime, $n\in\mathbb{Z}^+$
37947
Prove that if a number $n > 1$ is not prime, then it has a prime factor $\le \sqrt{n}$.
113359
Proving that $x^2 + 4$ is not divisible by $3$
2802
If $2^n+1$ is prime, why must $n$ be a power of $2$?
107725
Prove that if $n$ is not divisible by $5$, then $n^4 \equiv 1 \pmod{5}$
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.
35052
Prove that if $m^2 + n^2$ is divisible by $4$, then both $m$ and $n$ are even numbers.
26449
$|G|=24$ prove that $G$ is not simple
102889
If $2^n - 1$ is prime from some integer $n$, prove that n must also be prime.