$x^{2} \equiv-1\ \pmod p$ has a solution if and only if $p\equiv 1\ \pmod 4$
27287
$-1$ is a quadratic residue modulo $p$ if and only if $p\equiv 1\pmod{4}$
33469
Diophantine equation $ax + by = c$ has an integer solution $x_0, y_0$ if and only if $\gcd(a,b)|c$
73238
Show that $a^{q-1} \equiv 1 \pmod{pq}$
82571
Number of solutions of $x^2=1$ in $\mathbb{Z}/n\mathbb{Z}$
89590
The number of solutions of $ax^2+by^2\equiv 1\pmod{p}$ is $ p-\left(\frac{-ab}{p}\right)$
7085
Prove that if the equation $x^{2} \equiv a\pmod{pq}$ has any solutions, then it has four solutions.
21250
Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$
13904
$p = x^2 + xy + y^2$ if and only if $p \equiv 1 \text{ mod }3$?
71482
Solve $p_4(x) = x^4 −(2m + 4)x^2 + (m−2)^2 $such that $p_4$ is a product of two non-constant integer-coeficient polynomials
46028
Show that the solution for the Diophantine equation $x^2 - y^2 = N$ is unique if and only if $|N|$ or $\frac{|N|}{4}$, respectively, is $1$ or prime.
41890
How to get all solutions of $x^2 \equiv$ $a$ mod $p$?
75199
$a \equiv b \pmod n$ and $c\equiv d \pmod n$ implies $ac \equiv bd \pmod n$
38876
$r! \equiv (−1)^k \pmod p$
44033
Number theory lemma - $x^2 \equiv 1 \pmod p$ implies $x\equiv\pm 1\pmod p$
86594
If $x^2 \equiv y^2 \pmod {p^r}$, where $p$ is an odd prime not dividing $x$ or $y$ then $x \equiv \pm y \pmod {p^r}$
134450
Chinese Remainder Theorem $\ x\equiv m_k-2\pmod{m_k}$
122741
$x^2$ is congruent to $-1 \bmod p$