query: For which prime numbers $p$ there exist $x,y\in \Bbb{Z}$ such that $p=x^2+2y^2$?
positive_passages.docid: 20486
positive_passages.text: Numbers representable as $x^2 + 2y^2$
negative_passages.docid: 136856
negative_passages.text: Diophantine equation with a prime number
negative_passages.docid: 151985
negative_passages.text: If $p$ is a prime and $x,y \in \mathbb{Z}$, then $(x+y)^p \equiv x^p+y^p \pmod{p}$
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: 70463
negative_passages.text: 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 $0<|x|,|y|<\sqrt{p}$
negative_passages.docid: 46383
negative_passages.text: If $x$ divides $z$, $y$ divides $z$ and $x$ and $y$ are relatively prime, then $xy$ divides $z$.
negative_passages.docid: 50663
negative_passages.text: prove or disprove: For a prime number p, there is a positive integer n such that $p+1 $= $n^2$ if and only if $ p = 3.$
negative_passages.docid: 46028
negative_passages.text: Show that the solution for the Diophantine equation $x^2 - y^2 = N$ is unique if and only if $|N|$ or $\frac{|N|}{4}$, respectively, is $1$ or prime.
negative_passages.docid: 18053
negative_passages.text: Find all prime $p$ such that $x^2=-1$ has a solution in $\mathbb{Z}/p\mathbb{Z}$
negative_passages.docid: 71756
negative_passages.text: diophantine equation: $x^2 +y^2 = z^n$
negative_passages.docid: 123159
negative_passages.text: Solutions of the diophantine equation $x+y+z=xyz$?
negative_passages.docid: 69582
negative_passages.text: Is it true that: $a+bi$ is prime in $\mathbb{Z}[i]$ if and only if $a^2+b^2$ is prime in $\mathbb{Z}$
negative_passages.docid: 73515
negative_passages.text: Primes Number Theory
negative_passages.docid: 144814
negative_passages.text: If $p$ and $q$ are primes, which binomial coefficients $\binom{pq}{n}$ are divisible by $pq$?
negative_passages.docid: 344
negative_passages.text: Prove that if $p\in \mathbb{Z}$ is irreducible, then it is also prime.
negative_passages.docid: 45454
negative_passages.text: Pairs of $(p,q)$ such that $p^2 + 1 = 2 q^2$?
negative_passages.docid: 18302
negative_passages.text: List the primes for which the following system of linear equations DOES NOT have a solution in $\mathbb{Z}_p$