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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The 'prime logarithm'

Lately I've been thinking about the functional equation $$f(ab) = f(a) + f(b)$$ but not in the usual sense where continuity or differentiability are assumed. It's clear that $f(1) = 0$, by letting $a =...
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Regarding one form of potential primes

If we think of primes of the form $a^n-b^n,$ where $a,b,n$ are positive natural numbers and $a>b$, $(a-b)\mid (a^n-b^n)$, so $a-b$ must be $1$ and $n$ must be prime else $(a^r-b^r)\mid (a^n-b^n)$ ...
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To find whether $a$ is a prime number

I have been using this rule to determine whether a number is a prime number, but not how to prove it. Why it has to be $\sqrt{a}$? If $a$ is not divisible by all the prime numbers less than or ...
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What is a distinct prime?

I need to know what a distinct prime is, and what happens when you multiply two of them. How can I figure this out?
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How do I make this formula for the primes more concise?

The form I made for the $(n+1)^{th}$ prime $p_{n+1}$ is $\displaystyle1+\sum_{j=1}^{2p_n-1}\lfloor\frac{p_n!^j}{j!}\rfloor-\lfloor\frac{p_n!^j-1}{j!}\rfloor=p_{n+1}.$ Problem is, just like any ...
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Every even integer $n>2$ is a semiprime or sum of two semiprime numbers.

Progress: A slightly stronger version of the original assumption is this: Every even integer $n>2$ is a semiprime or sum of two even semiprime numbers. I was wondering as to how this ...
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Why Fibonacci(prime-1) or Fibonacci(prime+1) is divisible by that prime?

Why Fibonacci(prime-1) or Fibonacci(prime+1) is divisible by that prime and Fibonacci(nonprime-1) or Fibonacci(nonprime+1) is not divisible by that nonprime? Is there any elegant proof of that?
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How does the Riemann Hypothesis show the prime spectrum with zeros?

I learned that dependent on the Riemann Hypothesis $$d(x)=-\frac{1}{\pi}\sum_{p^n}\frac{\ln(p)}{p^{\frac{n}{2}}}\cos(x\ln(p^n))$$ has peaks converging at the real points $t$ where $\zeta(\frac{1}{2} + ...
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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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A conjecture about primes

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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Arithmetic sequence whose any five consecutive elements contain a prime

Consider an arithmetic sequence $\{11 + 13k : k\in\mathbb{N}\cup\{0\} \}$ Does this sequence contain five consecutive composites? If we look at some selections of five consec. elements: $$11, 24, 37, ...
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Proof Enquiry, Field of order $p^n$ [duplicate]

I want to prove that there exists an inclusion $\mathbb{F}_{p^a} \hookrightarrow \mathbb{F}_{p^b}$ iff $a \vert b$. Suppose that $a \vert b$, then $b =ac$ for some $c \in \mathbb{Z}$. Consider then ...
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Is there a name for these primes?

What is the name for primes $p$ where $2p-1$ is also a prime? $2p+1$ is a Sophie Germain prime. On average if $p$ is a primes how many primes of form $2p^n-1$ could we expect where $0<n<B$ ...
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Proving primality of $p$ without making any calculation involving $p$ directly

Wilson's Theorem states that a positive integer $p > 1$ is prime if and only if $(p-1)! \equiv -1 \pmod p$, showing a relationship between factorials and prime numbers. Finding it curious, today I ...
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what is Prime Gaps relationship with number 6?

Out of the 78499 prime number under 1 million. There are 32821 prime gaps (difference between two consecutive prime numbers) of a multiple 6. A bar chart of differences and frequency of occurrence ...
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Factorial and primorial twin primes

Factorial primes are are primes of the form $n! \pm 1$ and primorial primes are primes of the form $p\#\pm 1$, where $p\#$ is the product of all primes $\leq p$. To cite http://www.ams.org/journals/...
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Number of primes of a certain form

Let $p_n$ be the nth prime. Are there an infinite number of primes of the form $2p_n+1$? Is something known about questions like this?
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Is that true that all the prime numbers are of the form $6m \pm 1$?

Is that true that all the prime numbers are of the form $6m \pm 1$ ? If so, can you please provide an example? Thanks in advance.
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Possibly New Prime Conjecture

I was in the midst of proving a conjecture when I came across an observation that led me to forming a potentially new conjecture. The conjecture goes as follows: Any given sum of twin primes (...
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Do primes “behave” in this way?

Suppose that we choose some real number $\varepsilon >0$. Can we always find $n_0(\varepsilon) \in \mathbb N$ such that for every $n> n_0(\varepsilon)$ there is a prime number $p$ such that ...
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Show that any arithmetic progression contains a sequence of composites of arbitrary length

My question is inspired by this one: Arithmetic sequence whose any five consecutive elements contain a prime A more precise form: Let $(x_n)|_{n=1}^{\infty}$ be an arithmetic progression such that ...
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Fractals using just modulo operation

Let us calculate the remainder after division of $27$ by $10$. $27 \equiv 7 \pmod{10}$ We have $7$. So let's calculate the remainder after divison of $27$ by $7$. $ 27 \equiv 6 \pmod{7}$ Ok, so ...
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Formula for the number of numbers $\le n$ with same prime factors as $n$?

Is there a more concise formula for this? I threw this one together, $\sum_{j=1}^{n}(\lfloor\frac{n^{j}}{j}\rfloor-\lfloor\frac{n^{j}-1}{j}\rfloor)(\lfloor\frac{j^{n}}{n}\rfloor-\lfloor\frac{j^{n}-1}{...
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Highest ratio between consecutive prime numbers

Let $r = p_2/p_1$; where $p_1$, $p_2$ are consecutive prime numbers. What is the highest possible value of $r?$ Are there any consecutive prime numbers such that $r > 5/3$?
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Solutions to set of equations involving prime numbers

Is there a collection of distinct positive integers $(k_1, k_2, k_3, p_1, p_2, p_3)$ such that: $p_1, p_2, p_3$ are odd primes, and $k_1, k_2, k_3$ are odd $(k_1 + 2) p_1 = k_2 p_2$ and $(k_2 + 2) ...
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$\pi\left(\left(n+m\right)^2\right) - \pi\left(n^2\right) \ge 2 \cdot m$

Conjecture For $n \ge 1 $ , $m \ge 1$ $\pi\left(\left(n+m\right)^2\right) - \pi\left(n^2\right) \ge 2 \cdot m$ where $\pi\left(n\right)$ is the prime counting function . Does this conjecture ...
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How far to nearest/next prime?

Is there is metric to know how far we are from the nearest prime number. For example if my number is 38, then we are 3 numbers away from 41? If such a metric doesn't exist, is there an upper bound ...
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Express a prime $p$ as $p=a^2-2b^2$

Suppose $2$ is a quadratic residue modulo $p$ for an odd prime $p$. That is, there is an element $u$ such that $u^2 \equiv 2 \pmod{p}$. From here, can we prove that there exist integers $a$ and $b$ ...
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Do all primes occur in some sequence associated with the Collatz conjecture?

Let $f(n) = \begin{cases} n/2, & \text{if $n$ is even} \\ 3n+1, & \text{if $n$ is odd} \end{cases}$ For an arbitrary prime $p$ are there some start value $x_0$ such that $p = x_k$ for some ...
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Roots of the equation $x^2+1=0$ in $\Bbb Z/p^{n}\Bbb Z$

Let $p$ be an odd prime number and $n$ be a positive integer. I want to consider roots of the equation $x^{2}+1=0$ in the ring $\Bbb Z/p^{n}\Bbb Z$. Suppose $n=1$. Find a condition on $p$ such ...
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Prime numbers and arithmetic progressions

Whether there exist a polynomial $f$ such that for every $n$ there exist prime numbers $p_1, \ldots, p_n$, and an integer $b$ such that every $p_i$ and $b$ are less than $f(n)$ and $p_1×\ldots×p_n×b + ...
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Random binary array shows patterns around prime numbers

First post, so please let me know if I'm doing something wrong or if this question does not belong here. I have been toying with java to visualize an interesting 2D binary array I thought of today in ...
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Twin prime conjecture proof error

I am absolutely sure this is wrong but I can't find why. For every integer $n$ there exist a finite number of primes less than $n$. Take the set containing those primes and multiply them together to ...
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The GCD of a Univariate Integer-Valued Polynomial over a Set

Let $\mathcal{I}[X]$ denote the subring of $\mathbb{Q}[X]$ consisting of all integer-valued polynomials (i.e., $f(X)\in \mathbb{Q}[X]$ such that $f(k)\in\mathbb{Z}$ for all $k\in\mathbb{Z}$). For $f(...
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Proof of the following property about prime numbers

Is there a proper proof of the following property: Let $p$ be a prime number. The number of invertible elements in $\mathbb{Z}/p^n\mathbb{Z}$ is $(p-1)p^{n-1}$.
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$\mid-1+2-3+4-5+6-7+10-11+…+(p_{k}-1)-p_{k}\mid=k$? Where $p_{k}$ is the $k$-th prime.

I'm not sure if it's a Telescoping series but I tried the generating rule to prove and test the series but I'm not getting any insight and I got stuck. Here are few Examples: $$\mid-1+2-3+4-5+6-7\...
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Showing the infinitude of primes using the natural logarithm

I came across this proof in Proofs From the Book by Aigner and Ziegler. It uses the inequality $logx \leq \pi(x)+1$. (Here, we use natural logarithm) The proof starts with the inequality $log$ $x \...
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If $p > 3$ is prime, then $12 $ divides $p^2 - 1$

First up, I know there are a lot of similar questions with 24, not 12. So bare with me please :) What is the Question? Consider the following numbers of the form $p^2 - 1$ where $p$ is prime. $$5^2 ...
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Polynomial problem involving divisibility, prime numbers, monotony

Let $f$ be a polynomial function, with integer coefficients, strictly increasing on $\Bbb N$ such that $f(0)=1$. Show that it doesn't exist any arithmetic progression of natural numbers with ratio $r&...
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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} $ ...
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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 ...
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Can composite numbers be uniquely written as a sum of two squares?

Let $X = a^2 +b^2$ where all the terms are positive integers and $X$ is a composite number and $\gcd(a,b)=1$ . Do there exist positive integers $c$ and $d$ other than $a$ and $b$ such that $X = c^2+d^...
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Show that a positive integer $a>1$ is a perfect square (i.e., the square of an integer)…

if and only is in the prime decomposition of $a$ all the exponent are even integers. I don't understand what the question is asking. If I'm interpreting this correctly....any $a>1$ such as 9 would ...