query: Show that if $p \equiv 3$ mod $4$ is a prime in $\mathbb{Z}$, then $p$ is a prime in $\mathbb{Z}[i]$
positive_passages.docid: 19740
positive_passages.text: What's are all the prime elements in Gaussian integers $\mathbb{Z}[i]$
negative_passages.docid: 131971
negative_passages.text: If $k>0$ is a positive integer and $p$ is any prime, when is $\mathbb Z_p[\sqrt{k}] =\{a + b\sqrt k~|~a,b \in\mathbb Z_p\}$ a field.
negative_passages.docid: 50412
negative_passages.text: Prime elements in $\mathbb{Z}/n\mathbb{Z}$
negative_passages.docid: 140198
negative_passages.text: 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 $b^2\equiv a \mod p$.
negative_passages.docid: 130342
negative_passages.text: Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
negative_passages.docid: 32618
negative_passages.text: Suppose $p$ is a prime number such that $(p-1)/4$ and $(p+1)/2$ are also primes. Show that $p=13$
negative_passages.docid: 57669
negative_passages.text: A proof that $\mathbb{Z}_{p}$ is an integral domain if and only if $p$ is prime.
negative_passages.docid: 13317
negative_passages.text: show that $a^{12} \equiv 1 \mod 32760$ for each $a \in \mathbb{Z}$ such that $\text{gcd}(a, 32760) = 1$
negative_passages.docid: 157884
negative_passages.text: For any prime $p>3$ show that 3 divides $2p^2+1$
negative_passages.docid: 148671
negative_passages.text: Trying to factor $x^4-3$ over $\Bbb{F}_p$ with $p\equiv 3\mod{4}$
negative_passages.docid: 88036
negative_passages.text: Prove that if $p \equiv 3 \pmod{4}$ then $x^2+y^2 \not \equiv 0 \pmod{p}$
negative_passages.docid: 31064
negative_passages.text: Showing $x^8\equiv 16 \pmod{p}$ is solvable for all primes $p$
negative_passages.docid: 30374
negative_passages.text: If $n\in\mathbb N$ and $4^n+2^n+1$ is prime, prove that there exists an $m\in\mathbb N\cup\{0\}$ such that $n=3^m$.
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: 75318
negative_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: 145752
negative_passages.text: Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$
negative_passages.docid: 135586
negative_passages.text: Prove that the preimage of a prime ideal is also prime.