query: Prove that every prime $p$ of which $-1$ is a quadratic residue can be represented in the form $x^2+y^2$.
positive_passages.docid: 75318
positive_passages.text: How do you prove that a prime is the sum of two squares iff it is congruent to 1 mod 4?
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: 21629
negative_passages.text: Using Fermat's Little Theorem Prove if $p$ is prime, prove $1^p + 2^p + 3^p +...+(p-1)^p \equiv 0 \bmod{p}$
negative_passages.docid: 39135
negative_passages.text: Representation of integers by ternary quadratic form $x^2+y^2-z^2$
negative_passages.docid: 46292
negative_passages.text: Find all primes $p>2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$
negative_passages.docid: 100984
negative_passages.text: Prove that if prime $p$ divide $a_{2k}-2$, then $p$ divide also $a_{2k+1}-1$.
negative_passages.docid: 116686
negative_passages.text: $p=2^n+1$. Prove that every quadratic nonresidue modulo $p$ is a primitive root modulo $p$
negative_passages.docid: 119630
negative_passages.text: For an odd prime $p$, $\;p =1 \pmod 4,\; \;x^2+1 = 0 \pmod p$
negative_passages.docid: 121890
negative_passages.text: Proof that $ p $ | ${ p^n \choose k } $ for any prime $p$ and $ k < p^n$
negative_passages.docid: 27287
negative_passages.text: $-1$ is a quadratic residue modulo $p$ if and only if $p\equiv 1\pmod{4}$
negative_passages.docid: 5841
negative_passages.text: Proving prime $p$ divides $\binom{p}{k}$ for $k\in\{1,\ldots,p-1\}$
negative_passages.docid: 42376
negative_passages.text: Let $p$ be prime and $(\frac{-3}p)=1$. Prove that $p$ is of the form $p=a^2+3b^2$
negative_passages.docid: 138259
negative_passages.text: Proof or disproof that every integer can be written as the sum of a prime and a square.
negative_passages.docid: 37093
negative_passages.text: If p is an odd prime, prove that $1^2 \times 3^2 \times 5^2 \cdots  \times (p-2)^2 \equiv (-1)^{(p+1)/2}\pmod{p}$
negative_passages.docid: 44033
negative_passages.text: Number theory lemma - $x^2 \equiv 1 \pmod p$ implies $x\equiv\pm 1\pmod p$
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: 153862
negative_passages.text: A question about a primitive root mod $p=2^{2^k}+1$, where $p$ is prime.