Let $p > 2$ be prime. Show that there exist integers $a,b \geq 0 $ satisfying the congruence $a^2 + b^2 \equiv -1 $(mod $p$).
44019
Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
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 $0<|x|,|y|<\sqrt{p}$
130846
Prove or disprove: If $a \in \mathbb{Z}$, $a > 0$, and $p$ is a prime such that $p^4 \mid a^3$, then $p^2 \mid a$.
123441
Quadratic congruences problem $(x^2-a)(x^2-b)(x^2-ab)$ ≡ 0 (mod p)
102046
If all primes greater than some fixed value are congruent one modulo an integer, then said integer is equal two.
102234
Show $a^p \equiv a \mod 2p$
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 $b^2\equiv a \mod p$.
144851
For a prime $p \equiv 1$ or $3$ (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution.
11989
$ac\equiv bc \pmod{\!m}\!\!\iff\!\! a\equiv b \pmod{\!\!\frac{m}{(c,m)}}$ [Euclid's Lemma, congruence form]
75166
Find with proof all positive integers $n$ such that $2^n + 1 \equiv 0 \pmod{n}$
121717
Prove that $z^2 \equiv ab\pmod p$ is solvable if and only if both or neither of $x^2 \equiv a \pmod p$, $x^2 \equiv b \pmod p$ are solvable.
127816
Number theory proof mod p
73238
Show that $a^{q-1} \equiv 1 \pmod{pq}$
50781
$(n+1)$th prime $p_{n+1}$ less than or equal to $p_1p_2\dots p_n+1$
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 $a$
87267
Conjecture similar to Fermat's Theorem.
513
If n is an odd integer, show there exists a positive integer k such that 2^k mod n = 1.