query: Let $\pi$ be a prime element in $\Bbb{Z}[i]$. Then $N(\pi)=2$ or $N(\pi)=p$ s.t. $p$ is prime and $p\equiv 1\pmod 4$
positive_passages.docid: 154305
positive_passages.text: Let $\pi$ denote a prime element in $\mathbb Z[i], \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4$
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: 60804
negative_passages.text: About the property of $m$: if $n < m$ is co-prime to $m$, then $n$ is prime
negative_passages.docid: 73515
negative_passages.text: Primes Number Theory
negative_passages.docid: 57691
negative_passages.text: calculate (find the type of isomorphism) of $\mathbb Z_p/p\mathbb Z_p$ where p is prime
negative_passages.docid: 30134
negative_passages.text: For $a,n \in \Bbb N$, how to prove if $a^n-1$ is prime then $a=2$ and $n$ is prime?
negative_passages.docid: 137457
negative_passages.text: Use mathematical induction to prove that any integer $n\ge2$ is either a prime or a product of primes.
negative_passages.docid: 101345
negative_passages.text: Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n \mathbb Z$. Prove that either every element of $H$ is...
negative_passages.docid: 34954
negative_passages.text: If $11..11$($n$ times 1) is prime, prove that $n$ is also prime
negative_passages.docid: 125188
negative_passages.text: how to prove $m^{\phi(n)}+n^{\phi(m)}\equiv 1 \pmod{mn}$ where m and n are relatively prime?
negative_passages.docid: 146304
negative_passages.text: prove $(n)$ prime ideal of $\mathbb{Z}$ iff $n$ is prime or zero
negative_passages.docid: 145752
negative_passages.text: Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$
negative_passages.docid: 47186
negative_passages.text: $(1-\zeta_m)$ is a unit in $\mathbb{Z}[\zeta_m]$ if m contains at least two prime factors
negative_passages.docid: 32108
negative_passages.text: $ a\in \Bbb Z_n$ is invertible $\,\Rightarrow\gcd (a,n) =1$
negative_passages.docid: 66748
negative_passages.text: Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
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: 44857
negative_passages.text: Why the $(n+1)$th prime number cannot exceed the smallest prime divisor of $p_1\dotsc p_n + 1$