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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Are my calculations of a new constant similar to Mill's constant based on $\lfloor A^{2^{n}}\rfloor$ and Bertrand's postulate correct?

As Wikipedia explains in number theory, Mills' constant is defined as: "The smallest positive real number $A$ such that the floor function of the double exponential function $\lfloor A^{3^{n}}\...
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1answer
56 views

Divisibility of a summation by $p^2$

I try to use the hint of this problem but I could not. Any detailed answer will be appreciated! Let $p$ be a prime number which $p>3$, and $$a/b:=1+1/2+1/3+\cdots +1/(p-1).$$ How could we show ...
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52 views

Primality test for Thabit numbers of the first kind

Definition 1 Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$ , where $m$ and $x$ are nonnegative integers . Definition 2 Let $T_n=3 \cdot 2^n-...
12
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173 views

Have I discovered an analytic function allowing quick factorization?

So I have this apparently smooth, parametrized function: The function has a single parameter $ m $ and approaches infinity at every $x$ that divides $m$. It is then defined for real $x$ apart ...
4
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1answer
61 views

Asymptotic density of Zhang's primes

By this point, it is well known that Yitang Zhang's result implies for some $c$, there are infinitely many primes $p$ such that $p+c$ is also prime, and that the smallest such $c$ is less than $70,000,...
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1answer
31 views

Can a polynomial $p(x)$ generate only primes and 2-almost primes $\forall x \ge 0 \in \Bbb N$ or there is also a restriction for this to happen?

There is a simple demonstration to show that a polynomial of any degree can not generate only primes. Basically, if $p(x)=a_nx^n+...+a_1x^1+a_0$ is prime for every $x \in \Bbb N$ ($\Bbb Z$ would be ...
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2answers
61 views

Is there any formula to find prime numbers

I have found from a site this formula: Ok.I have found that this formula is correct.see the reason below. This part of formula is always $1$ or $zero$. it's zero when $(2m)!+1$ isn't dividable ...
2
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1answer
227 views

Show that the square of any prime number is the IONOF of an integer. [duplicate]

This has been asked here: IONOFs Problem Solving Solving a Word Problem relating to factorisation But they did not provide context or examples The ionof of an integer is the integer divided by ...
2
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2answers
45 views

Estimate for a prime product.

Is there a bound for $$\prod_{i=1}^{m}\Big(1-\frac{1}{p_i}\Big)$$ where $p_i$ is $i$th prime? What if $m=O(\log n)$?
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2answers
70 views

For any consecutive natural numbers $a_1,a_2$ are there infinitely many primes $p,q$ such that: $a_1<\dfrac{p}{q}<a_2$?

Progress: Let $a_1,a_2$ consecutive natural numbers; prove or disprove the infinitude of distinct prime pair $p,q$ which satidfies: $a_1<\dfrac{p}{q}<a_2$ The most challenging part of the ...
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3answers
398 views

Do Gödel numbers have a practical use?

Is there any example of Gödel numbers being actually used in practice? If so for what purpose?
3
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1answer
217 views

Conjecture about Rabin-Miller pseudo prime test

I tested the Rabin-Miller pseudo prime algorithm using a single test value and found that the number of false calls depends on the size of the number to test, reducing to a (conjectured) negligible ...
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0answers
20 views

There are at least: $\big\lfloor\sqrt{p_{n}}\big\rfloor(p_{n}-1)-|p_{n}-2n|+1$, primes less than $p_{n}^{2}$, where $p_{n}$ is the $n$-th prime?

There are at least: $\big\lfloor\sqrt{p_{n}}\big\rfloor(p_{n}-1)-|p_{n}-2n|+1$, primes less than $p_{n}^{2}$, where $p_{n}$ is the $n$-th prime? Is this true or false? If true, how does one prove it?...
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1answer
74 views

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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2answers
66 views

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 =...
2
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2answers
106 views

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) = \frac{...
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1answer
34 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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667 views

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 ...
3
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1answer
56 views

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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1answer
18 views

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 ...
4
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1answer
54 views

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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23 views

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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3answers
290 views

Show that the square of any prime number is the Factof of some integer

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
42 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 ...
4
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1answer
58 views

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 ...
2
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2answers
48 views

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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50 views

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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2answers
282 views

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 $(3,4),(4,3) \text{ and } (4,4)$...
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1answer
26 views

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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43 views

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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128 views

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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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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41 views

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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1answer
48 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
65 views

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
23 views

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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38 views

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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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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1answer
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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 ...
2
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1answer
89 views

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 ...
2
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2answers
108 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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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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1answer
74 views

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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28 views

$\mid-1+2-3+4-5+6-7+10-11+…+(p_{k}-1)-p_{k}\mid=k$? 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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691 views

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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49 views

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 ...