Let a ∈ Z and suppose that the congruence $x ^2 ≡ a \mod p$ can be solved for all primes p except possibly finitely many, then a is square .
59261
If $n\in\Bbb Z^+$ is not a square, prove exist infinitely many primes $p$ such that $\left(\frac{n}{p}\right)=-1$.
20161
Prove that the multiplicative inverse of $a$ modulo $m$ exists if and only if $a$ and $m$ are coprime.
108035
solving diophantine equations using congruences
144851
For a prime $p \equiv 1$ or $3$ (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution.
41890
How to get all solutions of $x^2 \equiv$ $a$ mod $p$?
100426
Let $a$ and $b$ be integers. Show that if $a \equiv b \pmod{p}$ for every prime $p$, then $a = b$.
75528
$x^2-x+1$ has a root $\!\bmod p\,$ for infinitely many primes $p$
33735
The square of an integer is congruent to 0 or 1 mod 4
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 $b^2\equiv a \mod p$.
102046
If all primes greater than some fixed value are congruent one modulo an integer, then said integer is equal two.
19071
solution set for congruence $x^2 \equiv 1 \mod m$
44019
Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
94917
Prove that for every prime $p$, there exists a solution to $(x^2-2)(x^2-6)(x^2-3) \equiv 0 \mod(p)$
84603
Prove that if $p$ is a prime such that $p^2+2$ is a prime then $p=3$.
60305
Proving the congruence $p^{q-1}+q^{p-1} \equiv 1 \pmod{pq}$
78638
Find out all solutions of the congruence $x^2 \equiv29 \mod 5$.
14809
Existence & uniqueness of solutions of linear congruences $\ cx\equiv b\pmod m$