Let $n > 1$ be a natural number and $p$ be a prime such that $p\mid n^3-1$ and $n\mid p-1$. Prove that $4p-3$ is a perfect square.
103360
Prove $4p-3$ is a square knowing that $n\mid p-1$ and $p\mid n^3-1$, $p$ prime
70308
Prove that if $p$ and $p+2$ are both prime integers greater than $3$, then $6$ is a factor of $p+1$.
42376
Let $p$ be prime and $(\frac{-3}p)=1$. Prove that $p$ is of the form $p=a^2+3b^2$
27170
$a\in \mathbb{N}$, $p$ prime, $a<p$ prove that $a\mid p+1\iff\exists\, b,c\in\Bbb N:\dfrac{a}{p}=\dfrac{1}{b}+\dfrac{1}{c}$
86285
Given positive integers $ x , y$ and that $3x^2 + x = 4y^2 +y$ Prove that $x-y$ is a perfect square
105564
If $n$ is a positive integer greater than 1 such that $3n+1$ is perfect square, then show that $n+1$ is the sum of three perfect squares.
66851
If $p,q$ are primes $(pq,a)=1$ and $p-1|q-1$ then prove $a^{q-1}\equiv 1\pmod{pq}$
14724
For primes $p≡3\pmod 4$, prove that $[(p−1)/2]!≡±1\pmod p$.
23830
Proof that if $p\equiv3\,\left(\mbox{mod 4}\right)$ then $p$ can't be written as a sum of two squares
33714
$p$ prime, $p\mid a^k \Rightarrow p^k\mid a^k$
40968
show if $n=4k+3$ is a prime and ${a^2+b^2} \equiv 0 \pmod n$ , then $a \equiv b \equiv 0 \pmod n$
142513
Prove that $2xy\mid x^2+y^2-x$ implies $x$ is a perfect square.
24820
Prove that if $n$ is a perfect square, then either $n \equiv 1 \pmod 4$, or $n \equiv 0 \pmod 4$.
126494
Prove that there doesn't exist any integer $x \ge 3$ such that $x^2-1$ is prime.
58935
If $p$ and $p^2+8$ are both prime numbers, then $p^3+4$ is prime.
94019
Find all $p$'s such that $p^4 + p^3 + p^2 + p +1$ is a perfect square.
131488
Show that $p^3+4$ is prime