query: $p$ is an odd prime of the form $p=x^2+2y^2$ iff $p\equiv_8$ $1$ or $3$
positive_passages.docid: 144851
positive_passages.text: For a prime $p \equiv 1$ or $3$ (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution.
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: 27063
negative_passages.text: Prove that if n is an odd positive integer , then $n^2\equiv1\mod8$
negative_passages.docid: 113994
negative_passages.text: Let $p$ be a prime of the form $3k+2$ that divides $a^2+ab+b^2$ for some integers $a,b$. Prove that $a,b$ are both divisible by $p$.
negative_passages.docid: 127901
negative_passages.text: If $x^2+y^2 \equiv 0\pmod{p}$, then $p \equiv 1 \pmod{4}$
negative_passages.docid: 54359
negative_passages.text: Odd Primes Problem Proof
negative_passages.docid: 146512
negative_passages.text: Prove that for every prime number $p>3$, $4p^2+1$ can be written as the sum of three square numbers
negative_passages.docid: 123033
negative_passages.text: Showing a number $n$ is prime if $(n-2)! \equiv 1 \pmod n$
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: 78917
negative_passages.text: Show that for any prime numbers $p,q,r$, one has $p^2+q^2 \ne r^2$.
negative_passages.docid: 92416
negative_passages.text: About a sum of Legendre symbols associated to an odd prime
negative_passages.docid: 76644
negative_passages.text: For any odd prime $p$, a quadratic is solvable mod $p^2$ if it is solvable mod $p$, and $p$ does not divide the discriminant and leading coefficient.
negative_passages.docid: 156607
negative_passages.text: Show that $L_p =\frac{1^p+2^p+3^p+...+(p-1)^p}{p}$ is an integer and generalize the integer for all odd integers p
negative_passages.docid: 38805
negative_passages.text: Prove that $x^{2} \equiv -1 \pmod p$ has no solutions if prime $p \equiv 3 \pmod 4$.
negative_passages.docid: 11564
negative_passages.text: $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})=\mathbb{Q}(\sqrt{p}\sqrt[3]{q})$ for distinct prime $p,q$
negative_passages.docid: 12365
negative_passages.text: Show that every prime $p>3$ is either of the form $6n+1$ or of the form $6n+5$
negative_passages.docid: 88036
negative_passages.text: Prove that if $p \equiv 3 \pmod{4}$ then $x^2+y^2 \not \equiv 0 \pmod{p}$