For $a\in \mathbb{Z}$, $x^2 = a$ has a solution $x\in \mathbb{Z}$ IFF $x^2 \equiv a \pmod n$ for all $n \in \mathbb{Z}$
38911
Squares in $\mathbb{Z}_p$ that are not squares in $\mathbb{Z}$
20834
Integer solutions to $c=a^2-b^2$
5297
If G is a finite abelian group and $a_1,...,a_n$ are all its elements, show that $x=a_1a_2a_3...a_n$must satisfy $x^2=e$.
144851
For a prime $p \equiv 1$ or $3$ (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution.
35664
If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic
152451
REVISITED $^2$: Does a solution in $\mathbb{R}^n$ imply a solution in $\mathbb{Q}^n$?
79219
Prove that $a^{2^n} \equiv 1 \pmod{2^{n+2}},$ where $a$ is an odd integer.
139
Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3)
52672
Solution of $\frac{a}{b}=\frac{a'}{b'}$ if $a,b,a',b' \in \mathbb{N}$
33314
How many non-negative integer solutions does $x_1+x_2+\cdots+x_n=A$ have?
125868
Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.
95105
Solutions in complex number field, instead of $\mathbb{N}$, to Fermat's Last Theorem
41890
How to get all solutions of $x^2 \equiv$ $a$ mod $p$?
152167
Prove that if $x,y,z$ are integers such that $x^2+y^2=z^2$, then $xyz \equiv o \pmod{60}$.
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.
135986
Show if $a \equiv b \pmod{m}$ then $ac \equiv bc\pmod{mc}$
155743
Why does the equation $x^2\equiv2 \pmod 5$ have no solutions?