query: There exists an integer $x$ s.t $x^2\equiv -3 \,\text{mod}\, p$ iff $p=3k+1$, $k\in\mathbb{Z}$
positive_passages.docid: 146851
positive_passages.text: Prove that $-3$ is a quadratic residue mod an odd prime $>3$ if and only if $p$ is of the form of $6n+1$
negative_passages.docid: 5978
negative_passages.text: Prove that there exists only one prime number of the form $p^2−1$ where $p≥2$ is an integer
negative_passages.docid: 135465
negative_passages.text: Group theory proof of existence of a solution to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$
negative_passages.docid: 6332
negative_passages.text: If $p\equiv1\mod{4}$, then $p=a^{2}+b^{2}$ where a and b are uniquely determined
negative_passages.docid: 26301
negative_passages.text: Prove that $x^3 \equiv x \bmod 6$ for all integers $x$
negative_passages.docid: 157884
negative_passages.text: For any prime $p>3$ show that 3 divides $2p^2+1$
negative_passages.docid: 82921
negative_passages.text: Uniqueness of $k$th root mod $m$ if $(k, \phi(m)) = 1$.
negative_passages.docid: 87267
negative_passages.text: Conjecture similar to Fermat's Theorem.
negative_passages.docid: 63478
negative_passages.text: Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$?
negative_passages.docid: 23830
negative_passages.text: Proof that if $p\equiv3\,\left(\mbox{mod 4}\right)$ then $p$ can't be written as a sum of two squares
negative_passages.docid: 122932
negative_passages.text: When $(\mathbb{Q}(\zeta_n) : \mathbb{Q}) = 2$?
negative_passages.docid: 34154
negative_passages.text: Prove the statement $\forall x\in\mathbb N (x > 1\to\exists k\in\mathbb N\exists m \in\mathbb N (m \equiv 1 (\text{mod }2) \wedge x = 2^km))$.
negative_passages.docid: 105442
negative_passages.text: Show that any prime divisor of $x^4+x^3+x^2+x+1$, with $x\in\mathbb{N}$, is $5$ or $1$ mod $5$
negative_passages.docid: 84017
negative_passages.text: Is $\sqrt[3]{2}$ contained in $\mathbb{Q}(\zeta_n)$?
negative_passages.docid: 136279
negative_passages.text: Infinitely many primes $\equiv 3 \mod 4$
negative_passages.docid: 111505
negative_passages.text: Proof $\binom{p-1}{k}\equiv (-1)^k \mod{p}$
negative_passages.docid: 127945
negative_passages.text: Number of solutions to the congruence $x^q \equiv 1 \mod p$.