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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Origin of period function model of primes

There is a web page attributed to Omar Pol, "Sobre el patrón de los números primos: Determinación geométrica de los números primos y perfectos." ("On the pattern of primes: Geometric Determination of ...
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1answer
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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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2answers
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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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1answer
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Numbers $m = pq^4$ ($p,q$ are distinct primes) for which $m$ divided by the number of its factors is an integer

The $\operatorname{Ionof}$ (Integer on number of factors) of an integer is the integer divided by the number of factors it has. For example, $18$ has $6$ factors so $\operatorname{Ionof}(18) = ...
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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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1answer
23 views

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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1answer
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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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1answer
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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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Prove that if $p \mid a-b$ then $p^{n+1} \mid a^{p^n}-b^{p^n}$

I need help with the following problem, I don't know how to continue. Let $p$ be a prime. Prove that if $p \mid a-b$ then: $$p^{n+1} \mid a^{p^n}-b^{p^n}$$ At first I thougt the following: $$p \mid ...
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+50

A conjecture about the prime function $p_n$

While testing my system Zet for computational mathematics I find possible relations now and then. The latest is: Conjecture: For all $(m,n)\in\mathbb Z_+^2$ except $(3,4),(4,3) \text{ and } ...
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Interesting and unusual word problem with prime numbers and factors

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with prime numbers, but other than that, the textbook gave no hints really and ...
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1answer
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Question related to division in $\mathbb{N}$ [on hold]

Let $p,a,b\in\mathbb{N}$. Prove that if $p$ does not divide $a$ or $b$ but divides $a\cdot b$, then $p$ cannot be a prime number.
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1answer
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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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A Product of Primes [on hold]

Anyone ever seen a function like this: F(1)=1st prime, F(2)=2nd prime * 1st prime, F(n)=nth prime * F(n-1)?
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1answer
41 views

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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2answers
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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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How to prove that $p$ divides $a^p -a$ for every integer $a$. [closed]

How to prove this Fermat's little theorem: $p$ divides $a^p -a$ for every integer $a$.
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3answers
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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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3answers
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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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1answer
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How to solve difficult positive integers and co-prime word problem?

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with derivative of algebra and prime numbers, which yields the shortest, ...
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1answer
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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 ...
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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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1answer
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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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1answer
47 views

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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1answer
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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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793 views

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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1answer
58 views

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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1answer
30 views

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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1answer
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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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2answers
105 views

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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1answer
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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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1answer
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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 ...