If $p^q - 1$ is a prime, then $p=2$ and $q$ is a prime
73095
Foundational proof for Mersenne primes
5978
Prove that there exists only one prime number of the form $p^2−1$ where $p≥2$ is an integer
158161
$\mathbb{Q}(\sqrt p_{1}+\sqrt p_{2})=\mathbb{Q}(\sqrt p_{1},\sqrt p_{2})$ for $p_{1},p_{2}$ primes.
10403
Let $p$ be a prime integer. Show that for each $a ∈ GF(p)$ there exist elements $b$ and $c$ of $GF(p)$ satisfying $a = b^2 + c^2$.
140830
Show $\,p\mid 9^q -1 \implies q\mid p-1$ for odd primes $p$ and $q$
64148
If $p$ and $q$ are distinct primes and $a$ be any integer then $a^{pq} -a^q -a^p +a$ is divisible by $pq$.
153862
A question about a primitive root mod $p=2^{2^k}+1$, where $p$ is prime.
13478
Prove that if $a^p-b^p$ is divisible by $p$, then it is also divisible by $p^2$
130342
Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
119282
If $a_1,a_2,\ldots, a_n$ are distinct primes, and $a_1=2$, and $n>1$, then $a_1a_2\cdots a_n+1$ is of the form $4k+3$.
147777
If $P$ and $Q$ are distinct primes, how to prove that $\sqrt{PQ}$ is irrational?
75685
Prove that there is prime number and natural so $n\times q$ digits are $1$.
44600
How can I prove: if $p $ is prime and $n>1$, then $ p^{\frac1n} $ is irrational?
19700
Prove that if $a$ and $b$ are relatively prime, then $\gcd(a+b, a-b) = 1$ or $2$
34323
Prove that $a^{(p-1)/2} \equiv 1$ (mod p) and $a^{(p-1)/2} \equiv -1 \pmod p$
834
If $p$ is a prime and $p \mid ab$, then $p \mid a$ or $p \mid b$.
140294
prove that $n >2$ and $n$ is a prime then $n$ is odd