# Tagged Questions

Prime numbers are natural numbers greater than 1 not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

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### Is 2 is prime in $\mathbb{ Z}_6$?

Prove that $2$ is prime element in $\mathbb{ Z}_6$? I have proved it using Caleys Table, but can someone suggest a theoretical method ?
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### As $n$ grows sufficiently larger, $\pi(n)<\pi_{1}(n)$, where $\pi(n)$ and $\pi_{1}(n)$ is the number of prime and semiprime $\leq{n}$, respectively

From $P_{12}=37$ the number of semiprime(s) appears to be higher than the number of prime(s). Though I couldn't check for a higher $n\geq{500}$ for several limitations, I could really use any proof or ...
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### Galois Group of Splitting Field, $S_4$

I've shown that the polynomial $x^4+px+p \in \mathbb{Q}[x]$, where $p$ is prime, is irreducible by Eisenstein's criterion. However, it remains to be shown that the Galois group of the splitting field ...
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### On a certain prime structure.

It is unknown whether there are infinite primes $p$ where $2p-1$ is also a prime. Is it known there are only finitely many primes $p$ such that both $q$ and $2p-1$ are primes where $p-1=2aq$ for any ...
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### Range to look for first $N$ prime numbers.

What range of numbers $[2, X]$ should I search, to be absolutely sure that I would get exactly or more than $N$ prime numbers within that range? Any formula for $X$?
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### When is a prime $p$ a quadratic residue modulo $3$?

Simple. When $p \equiv 1 \pmod 3$, it is a quadratic residue, and when $p \equiv -1 \pmod 3$ it is not a residue. So can we have a nice expression for the Legendre symbol $\left(\frac{p}{3}\right)$? ...
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### Proof — Infinitely many primes of the form $4k + 3$ — origin of $4(p_1…p_k - 1) + 3$

I know there are sundry questions — like this pdf — and this (10.) Prove that any positive integer of the form $4k + 3$ must have a prime factor of the same form. Because $4k + 3 = 2(2k + 1) + 1$, ...
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Let $p_n$ be the $nth$ prime and define $p_n^{(m)}$ by $p_n^{(1)}=p_n$ and $p_n^{(m+1)}=p_{p_n^{(m)}}$: $p_n^{(2)}=p_{p_n}$, $\;p_n^{(3)}=p_{p_{p_n}}$ and so far... For some coprime numbers $a,b$, ...
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### Is it true that a cycle with a period of 29 hours over 24 hours leads to a non-recurring pattern and how to prove it?

The default 'reset time' for Internet Information Services is 29 hours. The reason for this is that 'Wade [person on the team who developed the setting] suggested 29 hours for the simple reason ...
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### Uniquely identify any finite subset of an infinite set

Let $U$ be an unbounded subset of $\mathbb{N}$. Let $D = \mathcal{P}_{<\omega}(U)$ (the set of all finite subsets of $U$). Let $f$ be an injection such that: $f: D \rightarrow \mathbb{N}$ ...
### Sum of products of $(1 - 1/p)$
Let $\pi(n)$ denote the number of primes not greater than $n$, and $p_k$ the $k$th prime, so that $p_{\pi(n)}$ denotes the largest prime not greater than $n$. I'm interested in the value of the ...