Let p be an odd prime, where a,b are integers such that pł a and płb. Show that the congruence (x^2-a)(x^2-b)(x^2-ab) ≡ 0(modp) is always solvable.
123441
Quadratic congruences problem $(x^2-a)(x^2-b)(x^2-ab)$ ≡ 0 (mod p)
127945
Number of solutions to the congruence $x^q \equiv 1 \mod p$.
81601
Prove that if $x$ is odd then $x^2 -1$ is divisible by $8$.
19071
solution set for congruence $x^2 \equiv 1 \mod m$
124995
$p\equiv 1,2,3\text{ mod }8 \Rightarrow p=a^2+2b^2\text{ for some $a,b\in\mathbb{Z}$.}$
135465
Group theory proof of existence of a solution to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$
18245
How to check if a system of congruence is solvable?
10403
Let $p$ be a prime integer. Show that for each $a ∈ GF(p)$ there exist elements $b$ and $c$ of $GF(p)$ satisfying $a = b^2 + c^2$.
7414
Prove that odd perfect square is congruent to $1$ modulo $8$
108237
Show $a^p \equiv b^p \mod p^2$
130300
Proof that every quadratic residue has two roots, modulo a prime
37093
If p is an odd prime, prove that $1^2 \times 3^2 \times 5^2 \cdots  \times (p-2)^2 \equiv (-1)^{(p+1)/2}\pmod{p}$
140198
Let $a$ be a quadratic residue modulo $p$. Prove that the number $b\equiv a^\frac{p+1}{4} \mod p$ has the property that $b^2\equiv a \mod p$.
78638
Find out all solutions of the congruence $x^2 \equiv29 \mod 5$.
70463
Let $p$ be prime and $a$ be a natural not divisible by $p$. Then, $\exists$ integers $x,y$ such that $ax\equiv y \pmod{p}$ with $0<|x|,|y|<\sqrt{p}$
130342
Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
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.