If $p$ is a prime of the form $4k+3$ and $n$ is any integer, then $p$ doesn't divide $n^2+1$.
38805
Prove that $x^{2} \equiv -1 \pmod p$ has no solutions if prime $p \equiv 3 \pmod 4$.
26336
Prove that $n(n^2 - 1)(n + 2)$ is divisible by $4$ for any integer $n$
50413
Proof: if $p$ is prime, and $0<k<p$ then $p$ divides $\binom pk$
142584
Confusion in proof that primes $p = 4k + 1$ are uniquely the sums of two squares
78343
$4x^2+1$ factors only into $4y+1$ primes
50596
$p_1p_2...p_n +1$ is not square for $p_k = k$'th prime
27170
$a\in \mathbb{N}$, $p$ prime, $a<p$ prove that $a\mid p+1\iff\exists\, b,c\in\Bbb N:\dfrac{a}{p}=\dfrac{1}{b}+\dfrac{1}{c}$
100984
Prove that if prime $p$ divide $a_{2k}-2$, then $p$ divide also $a_{2k+1}-1$.
58935
If $p$ and $p^2+8$ are both prime numbers, then $p^3+4$ is prime.
140405
Show that there does not exist any $n$ for which $a^n=a$ forall $a\in \Bbb Z_m$ where $m$ is divisible by the square of some prime.
126494
Prove that there doesn't exist any integer $x \ge 3$ such that $x^2-1$ is prime.
138941
If p $\equiv$ 3 (mod 4) with p prime, prove -1 is a non-quadratic residue modulo p.
115758
Remainder of division when $p$ is not prime
64148
If $p$ and $q$ are distinct primes and $a$ be any integer then $a^{pq} -a^q -a^p +a$ is divisible by $pq$.
50781
$(n+1)$th prime $p_{n+1}$ less than or equal to $p_1p_2\dots p_n+1$
50663
prove or disprove: For a prime number p, there is a positive integer n such that $p+1 $= $n^2$ if and only if $ p = 3.$
5841
Proving prime $p$ divides $\binom{p}{k}$ for $k\in\{1,\ldots,p-1\}$