query: How dense are primes congruent to 1 and 3 (mod 4)?
positive_passages.docid: 157766
positive_passages.text: Are primes (ignoring $2$) equally likely to be $1~\text{or}~3\pmod 4$?
negative_passages.docid: 9043
negative_passages.text: finding infinetely many primes in three different forms of prime numbers below
negative_passages.docid: 4972
negative_passages.text: Mod of numbers with large exponents [modular order reduction]
negative_passages.docid: 57654
negative_passages.text: Congruent Modulo $n$: definition
negative_passages.docid: 127816
negative_passages.text: Number theory proof mod p
negative_passages.docid: 16459
negative_passages.text: Limit of ratios of numbers with $m$ factors and primes
negative_passages.docid: 107381
negative_passages.text: Understanding the proof of infinitely many primes $\equiv 3\pmod{\!4}$
negative_passages.docid: 69830
negative_passages.text: How often is $1+\prod_{k=1}^n p_k$ not prime?
negative_passages.docid: 20715
negative_passages.text: 31,331,3331, 33331,333331,3333331,33333331 are prime
negative_passages.docid: 126725
negative_passages.text: If $p$ is prime and $p$ $\equiv$ $1$ (mod 4), then the congruence $x^2$ $\equiv$ $-1$ (mod $p$) has two incongruent solutions...
negative_passages.docid: 41925
negative_passages.text: What is the distribution of primes modulo $n$?
negative_passages.docid: 121831
negative_passages.text: Proving that rational numbers are dense
negative_passages.docid: 158563
negative_passages.text: If $m^4+4^n$ is prime, then $m=n=1$ or $m$ is odd and $n$ even
negative_passages.docid: 139
negative_passages.text: Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3)
negative_passages.docid: 117516
negative_passages.text: Prime trios of the form $p$,$p+2$,$p+4$
negative_passages.docid: 71110
negative_passages.text: Show that we cannot have a prime triplet of the form $p$, $p + 2$, $p + 4$ for $p >3$
negative_passages.docid: 73515
negative_passages.text: Primes Number Theory