For a odd prime $p, \exists$ nonzero $a,b$ such that $a^2 + ab + b^2 \equiv 0 \pmod{p} \implies \exists x,y $ such that $x^2 + xy + y^2 = p$?
95162
For a odd prime $p, \exists a,b$ such that $a^2 + ab + b^2 \equiv 0 \pmod{p} \iff \exists x,y $ such taht $x^2 + xy + y^2 = p$?
13904
$p = x^2 + xy + y^2$ if and only if $p \equiv 1 \text{ mod }3$?
82808
Do there exist any odd prime powers that can be represented as $n^4+4^n$?
45454
Pairs of $(p,q)$ such that $p^2 + 1 = 2 q^2$?
121658
$x^2+y^2-1$ irreducible in $\mathbb{Q}[x,y]$?
37093
If p is an odd prime, prove that $1^2 \times 3^2 \times 5^2 \cdots  \times (p-2)^2 \equiv (-1)^{(p+1)/2}\pmod{p}$
77806
Proving if $a$ is not a multiple of prime $p$, then an integer $b$ exists such that $p^b - 1$ is a multiple of $a$
84603
Prove that if $p$ is a prime such that $p^2+2$ is a prime then $p=3$.
60780
Help needed in understanding proof: Every odd prime $p$ has exactly $(p-1)/2$ quadratic residues and $(p-1)/2$ quadratic nonresidues.
59442
Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$?
18053
Find all prime $p$ such that $x^2=-1$ has a solution in $\mathbb{Z}/p\mathbb{Z}$
10403
Let $p$ be a prime integer. Show that for each $a ∈ GF(p)$ there exist elements $b$ and $c$ of $GF(p)$ satisfying $a = b^2 + c^2$.
151701
Does there exist an $a_0$ such that the sequence $a_{n+1} = 2a_n + 1$ is prime for all $n \ge 0$?
44033
Number theory lemma - $x^2 \equiv 1 \pmod p$ implies $x\equiv\pm 1\pmod p$
38805
Prove that $x^{2} \equiv -1 \pmod p$ has no solutions if prime $p \equiv 3 \pmod 4$.
68790
Do we have $K(\beta)=K(\beta^2)$ for field extension of odd degree?
130342
Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.