query: Prove that the prime factors $p$ of $n=x^2+y^2$ of the form $p=4m+3$ can only have even exponents.
positive_passages.docid: 146241
positive_passages.text: If $a \in \Bbb Z$ is the sum of two squares then $a$ can't be written in which of the following forms?
negative_passages.docid: 52530
negative_passages.text: There exists a prime congruent to $m$ mod $p$
negative_passages.docid: 40968
negative_passages.text: show if $n=4k+3$ is a prime and ${a^2+b^2} \equiv 0 \pmod n$ , then $a \equiv b \equiv 0 \pmod n$
negative_passages.docid: 119051
negative_passages.text: Show that only finitely many positive integers $n$ have the property that for all $m$ for which $1<m<n$ and $\text{gcd}(m,n) = 1$, $m$ is prime.
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: 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: 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: 131488
negative_passages.text: Show that $p^3+4$ is prime
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: 38727
negative_passages.text: Modular Exponentiation and Primes
negative_passages.docid: 71788
negative_passages.text: Prove existence of multiplicative inverse modulo p when p is prime
negative_passages.docid: 106111
negative_passages.text: Understanding that there are infinitely many primes of form $4n+3$
negative_passages.docid: 57275
negative_passages.text: On the primality of integers of the form $p^2+k$
negative_passages.docid: 145041
negative_passages.text: Given that $p$ is a prime and $p\mid a^n$, prove that $p^n\mid a^n$.
negative_passages.docid: 100558
negative_passages.text: Prove that for any even positive integer $n$, $n^2-1 \mid 2^{n!}-1$
negative_passages.docid: 70308
negative_passages.text: Prove that if $p$ and $p+2$ are both prime integers greater than $3$, then $6$ is a factor of $p+1$.
negative_passages.docid: 75206
negative_passages.text: If $a$, $a+2$ and $a+4$ are prime numbers then, how can one prove that there is only one solution for $a$?