Prime numbers are natural numbers 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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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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253 views

Algorithm to find primes up to $n$ in $O\left(\frac{n}{\log n}\right)$?

Consider the problem of given an integer $n$, generating a list of the primes not greater than $n$. An optimized version of the Sieve of Eratosthenes can do such task in $O(n)$, while the more modern ...
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498 views

Sorting of prime gaps

Let $g_i $ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we re-arrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$ the gaps are arranged from smallest to largest we have a new sequence ...
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445 views

Divergence of the Derivative of the Prime Counting Function

On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written $$ \pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1} $$ with $ \operatorname{R}(z) = ...
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What other prime numbers have been ruled out as counterexamples to the Feit-Thompson conjecture?

Given distinct primes $p$ and $q$, $$\frac{p^q - 1}{p - 1}$$ is never a divisor of $$\frac{q^p - 1}{q - 1}.$$ Or so we believe. If $p = 2$, then it's clear that no odd prime $q$ can make a ...
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65 views

Consecutive prime numerators of harmonic numbers?

Let $$\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\frac{a}{b}$$ and let $a$ and $b$ are coprime, $h_{n}=a$. $h_{n}$ is prime for ...
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106 views

On the prime-generating polynomial $m^2+m+234505015943235329417$

In 2009, J. Waldvogel and Peter Leikauf found the remarkable Euler-like polynomial, $$F(m)=m^2+m+234505015943235329417$$ which is prime for $m=0\to20$, but composite for $m=21$. Define, ...
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145 views

Product of primes mod n

Let $n$ be an odd composite number. I'm trying to compute $$ f(n)=\prod_{n/2<p<n}p\pmod n $$ where $p$ ranges over the primes in the indicated region. Can this be done (significantly) faster ...
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206 views

Continued fraction with prime reciprocal entries

We know that the reciprocals of the primes form a divergent series. We also know that a necessary and sufficient condition for a continued fraction to converge is that its entries diverge as a series. ...
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164 views

Curious representation of primes

I found the following problem on the internet, and my initial intuition turned out to be entirely incorrect. The question asked what is the smallest prime $r$ that does not have a representation of ...
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413 views

Cramér's Model - “The Prime Numbers and Their Distribution” - Part 1

I was reading "The Prime Numbers and Their Distribution" by Gérald Tenenbaum and Michel Mendès France, the section about Cramér's Model, and I couldn't prove a couple of results. I would like to start ...
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368 views

A question about the divisibility of sum of 2 consecutive primes.

Well as I was curious about the sum of $2$ consecutive primes, after proving that the sum for the odd primes always has at least 3 prime divisors, I came up with this question: Find the least ...
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239 views

An infinitude of “congruence condition” primes?

Background: Several special classes of primes can be written as primes that satisfy some additional constraint $f(p)\equiv 0\pmod p$; for instance, Wilson primes are congruence primes with ...
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592 views

Understanding Ramanujan's approach in his proof of Bertrand's Postulate

I've been reading through Ramanujan's proof of Betrand's Postulate and I'm not clear why he didn't state his proof in terms of $\varphi(2x) - \varphi(x)$ What would be wrong with this approach for ...
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333 views

Showing that the Prime Number Theorem is Plausible.

I have started to work through the course notes titled "Integers, Polynomials and Finite Fields" by Kenneth Davidson to keep me busy this summer, and there is a question in here This is an ...
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457 views

Why does this identity equal the number of primes?

Can someone explain why this identity gives the number of primes? I don't understand it. $D_{0,a}(n) = 1$ $D_{k,a}(n) = \displaystyle\sum_{j=1}^{k} \binom{k}{j}\sum_{m=a+1}^{\lfloor ...
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Weak version of Fortune's conjecture

Let $p\#=2\cdot3\cdot5\cdots p$ denote the primorial and $N(x)$ the smallest prime greater than or equal to $x$. Then Fortune's conjecture is that $N(p\#+2)-p\#$ is prime for all $p$. (Heuristic: to ...
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Two (strictly related) proofs by induction of inequalities.

This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the ...
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118 views

Given the first $n$ primes, find the least common multiple of $p_1 - 1$, $p_2 - 1$, …, $p_n - 1$

Given the first $n$ primes, we can label the $k$th prime as $p_k$. So, what is the least common multiple(LCM) of {$p_1 - 1$, $p_2 - 1$, $p_3 - 1$, ..., $p_n-1$}? In other words, if we subtract $1$ ...
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134 views

Primality of $n! +1$

I came across with a problem where I was required to examine primality of $n! +1$ (17! + 1 was the actual number). Although Wilson's Theorem could be manipulated for determining primality of $n! + ...
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195 views

Intuition behind Erdős proof of the infinitude of prime numbers

Suppose by contradiction that there are finitely many primes, namely $p_1, p_2,...,p_k$, where $k$ is a natural number. Now consider another natural number $n$, and all natural numbers $m \leq n$. ...
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For primes $P_1$ and $P_2$, exists a prime $P_3$ that both $P_i + 6P_3$ is a prime

I was thinking about twin primes and I came to ask this question: If we have two distinct primes $P_1$ and $P_2$ which are both greater than $3$, then does there always exist a prime $P_3$ such that ...
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123 views

Is this a recurrence for the characteristic sequence of composite numbers?

The characteristic sequence of composite numbers is equal to 1 if $n$ is not a prime number and equal to 0 if $n$ is a prime number, starting: $$1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1,...$$ where the ...
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258 views

Why are minima of $(k \bmod 4)$-Prime $\zeta$ functions $|P_x(r,t)|$ more frequent for $\frac\pi2\leq t \leq \pi$?

I got these plots when I evaluate the sum of truncated $(k \bmod 4)$-Prime $\zeta$ function, i.e. $$ P_x(r,t)=P_{x;4,1}(-ir\cos t)+P_{x;4,3}(-ir\sin t)=\sum_{x\geq p\;\bmod\;4=3} p^{-ir\cos ...
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118 views

Asymptotics of the lower approximation of a pair of natural numbers by a coprime pair

When we are working, for instance, in combinatorics or graph theory, sometimes we can have the following situation. For each number $m$ from an infinite set $\mathbb M\subset\mathbb N$ we can ...
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How many numbers of the form $p_1^2 p_2 p_3$ are there less than $10^{15}$ for $p_1$, $p_2$, $p_3$ distinct primes?

Is there an easy way to compute the following question: How many numbers of the form $p_1^2 p_2 p_3$ are there less than $10^{15}$ for $p_1$, $p_2$, $p_3$ distinct primes? The only thing that ...
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196 views

Basic Olympiad number theory problem about prime numbers

I'm learning a basic lesson of number theory and get stuck with this : Find all positive integers $n$ and prime numbers $p$ such that $n^p+3^p$ is a perfect square. I found that if $p \ge 3$, then ...
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Differences of Prime Numbers

Let $a<b<c$ be primes such that $c-a$, $c-b$, and $b-a$ are also prime. It is rather simple to show that $(2,5,7)$ is the only triple that satisfies these conditions: Proof Sketch: The case ...
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Primes $p$ for which $2p \pm 1$ are also primes

Out of curiosity and trouble sleeping, I decided to look at the distribution of primes $p$ for which $2p \pm 1$ are also primes. I looked at the first 25,910,000 primes and counted the number of ...
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Where is The third Gisella prime?

A Gisella prime is a prime number obtained from concatenating the first $n$ natural numbers starting from $2$ and then replace each composite on that concatenation with the concatenation of its prime ...
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What is known about the sum $\sum\frac1{p^p}$ of reciprocals of primes raised to themselves?

Consider the following series: $$\sum_{p\in\mathcal{P}}\frac{1}{p^p}$$ where $\mathcal{P}$ is the set of all prime numbers: $\mathcal{P}=\{2,3,5,7,11,13,\ldots\}$. My question is: Is this a ...
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An Impossible Sequence of Prime Powers

Let $x_1,x_2,\ldots$ be a sequence of positive integers that satisfies the recurrence relation $$x_{n+1}=2x_n(x_n-1)+1$$ for all positive integers $n$. It seems impossible that every term in this ...
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Applying iterated function on the sum of the squares of the prime factors of $30$

Let $f(n)$ denote the sum of the squares of the prime factors of $n$ with multiplicity. For example, $f(60)=f(2\cdot2\cdot3\cdot5)=2^2+2^2+3^2+5^2=42$. Denote the iterated function ...
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The effect of roots of Dirichlet's $\beta$ function condenses to $\frac12\left(1+ie^{i2\pi\frac{p}4}\right)$

With the help of Raymond Manzoni and Greg Martin I was able to derive an explicit formula for the number of primes of the form $4n+3$ in terms of (sums of) sums of Riemann's $R$ functions over roots ...
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807 views

The prime number theorem and the nth prime

This is a much clearer restatement of an earlier question. In section 1.8 of Hardy & Wright, An Introduction to the Theory of Numbers, it is proved that the function inverse to $ x ⁄ \log⁡ x$ is ...
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Prime numbers, analysis of polylogarithms

Can any interesting results concering prime numbers be obtained using the analytic properties of the polylogarithm, similar to how analytic methods are used on the zeta function to obtain results ...
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118 views

Goldbach for certain classes of $n$

The Wiki article on the Goldbach conjecture (where $\#$ of ways even $n$ can be represented by prime additions is heareafter denoted $G(n)$) states that In 1975, Hugh Montgomery and Robert Charles ...
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A bound on the nth prime.

Is there any combinatorial argument to show that the nth prime $p_n = \mathcal{O}(n^k)$ for fixed $k$ ? There is a problem in the book by Apostol to find upper bounds on $p_n$, the Prime Number ...
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218 views

Relatives of Heegner numbers?

It is well known that Euler's lucky numbers are related to the Heegner numbers, where \begin{align} &n^2+n+p\\ \end{align} gives primes for $n=0,\dots,p-2$ if and only if its discriminant ...
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Does the set of $m \in Max(ord_n(k))$ for every $n$ without primitive roots contain a pair of primes $p_1+p_2=n$?

I have made the following observation: for those n even numbers that do not have primitive roots modulo n ,$Pr(n)$, the set $M(n)$ of those $k$ having a maximum multiplicative order $ord_n(k)$ ...
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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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To solve for $x,y,n$ in non-negative integers , $\dfrac{x!+y!}{n!}=p^n$ , $p$ a given prime

Let $p$ be a given prime , then how do we find non-negative integers $(x,y,n)$ $\space$ , such that $\dfrac{x!+y!}{n!}=p^n$ ?
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How to list the prime factorised natural numbers?

Today I set out to invent a two character numeral system designed to make factorization trivial. Indeed, it lets one factor non-trivial numbers with over thousand digits within 30 seconds per hand - ...
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141 views

Coprime multiplicative orders modulo infinitely many primes

Is it true that there are infinitely many primes $p$ such that the multiplicative orders of $2$ and $3$ are coprime $\pmod{p}$? By this I mean their order in $(\mathbb{Z}/p\mathbb{Z})^*$. If the ...
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Statement about Woodal primes.

A Woodal number is an integer of the form $n 2^{n}-1$. A Woodal prime is an integer that is both a prime and a Woodal number. Let $p$ be a prime of the form 1 mod 4. Then $p 2^{p} -1$ is never a ( ...
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Density of products of a certain set of primes

I have a set S of prime numbers and I would like to find the size (in some sense, ideally some nice asymptotic expression) of the set of positive integers which are the product of with all prime ...
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148 views

Number of primefactors in $ f(n,W) = \prod_{k=1}^W (p_k^n -1) \text{ where } p_k=Prime(k) $

I'm reviving an old fiddling, although I do not yet really see its benefit. Beginning with the eulerproduct for the zeta-function in the representation $\small \zeta(n)=\prod {1\over 1-p^{-n} } = ...
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278 views

Sums of Dirichlet-Characters over prime numbers (part 2)

This is kind of related to my previous question that was poorly stated because of misreading my own notes that I have taken on the papers I am currently reading, so no surprise that it eventually ...
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On the change $u=x^{1+\frac{1}{p_n}}$ in $\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx$, where $p_n$ is the nth prime number

In [1] Wikipedia say that for $\Re s>1$ the Riemann zeta function satisfies $$\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx,$$ where $\pi(x)$ is the prime counting function, and say too ...
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Elementary proof that finite sums of square roots of primes is irrational

It is relatively easy to show that if $p_1$, $p_2$ and $p_3$ are distinct primes then $\sqrt{p_1}+\sqrt{p_2}$ and $\sqrt{p_1}+\sqrt{p_2}+\sqrt{p_3}$ are irrational, but the only proof I can find that ...