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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Sheldon Cooper Primes

On the $73^{\text{rd}}$ episode of the Big Bang Theory, Dr. Sheldon Cooper, an astrophysicist portrayed by Jim Parsons $(1973 - \stackrel{\text{hopefully}}{2073})$ revealed his favorite number to be ...
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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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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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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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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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116 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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335 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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311 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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175 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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201 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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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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436 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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156 views

Euler's Totient function $\forall n\ge3$, if $(\frac{\varphi(n)}{2}+1)\ \mid\ n\ $ then $\frac{\varphi(n)}{2}+1$ is prime

While I was studying Euler's Totient function, $\varphi(n)$, I stumbled upon the marvelous book "Index to Mathematical Problems, 1980-1984" By Stanley Rabinowitz. In this page of the book (link 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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96 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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113 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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155 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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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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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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193 views

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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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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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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104 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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176 views

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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75 views

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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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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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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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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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 an infinite set S of prime numbers with relative density 0 (that is, $\lim_ns_n/p_n=\infty$ with $S=\{s_1,s_2,\ldots\}$ and $s_1 < s_2< \cdots$). I would like to find the size (in some ...
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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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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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A problem about $e^{2\pi i \alpha_1}+e^{2\pi i \alpha_2}+\cdots+e^{2\pi i \alpha_N}=0$

Let $\alpha_i\in [0,1),\; i\in \{1,\cdots,N\}$ for some positive integer $N$, such that $$e^{2\pi i \alpha_1}+e^{2\pi i \alpha_2}+\cdots+e^{2\pi i \alpha_N}=0$$ and if for any non-empty proper subset ...
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Given $h(\mathcal{O}_{\mathbb{Q}(\sqrt{d})}) = 1$, what is the longest possible run of inert primes in that ring?

Say $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ is a unique factorization domain. A couple of primes will ramify, the rest will either split or be inert. For example, in ...
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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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Is the Wikipedia article on the proof for Bertrand's Postulate correct?

I was checking out the Wikipedia article on the proof of Bertrand's Postulate. For Lemma 4, the argument is made for $x\ge 3, x\#<2^{2x-3}$ Here's the proof by induction: $n = 3$: $n\# = ...
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Primes as sum of squares.

If $p_{i}$ and $p_{j}$ are two primes of the form $4k+1$ , with $p_{j} > p_{i}$, show that if $p_{j} \neq$ sum of two squares $p_{i}$ is also not equal to sum of two squares. It is well ...
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More primes of the form $6k-1$ than $6k+1$

Let $a_n:= $ number of primes of the form $6k-1$ and $\le n$ and $b_n:= $ number of primes of the form $6k+1$ and $\le n$. I was playing with my computer and noticed that $a_n\ge b_n$ for ...
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Does this prime generating way generate all the prime numbers?

I've thought of the following algorithm to find the entire list of prime numbers: Take a prime number $p$ to your list. $1.$ Multiply all the numbers in your list and call the number you get ...
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Are $ut + 1$ and $ut + t + 1$ both prime for some t for any $u$?

Conjecture : For any natural number $u$, there is a natural number $t$ such that $ut + 1$ and $ut + t + 1$ are both prime. So we get a solution of the equation $$au - b(u+1) = -1$$ with prime ...
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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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Logical consequence of Euclid's theorem

Are there any far reaching non-trivial consequences of Euclid's infinitude of primes where theorems make use of it? Wikipedia does not have the list of applications of this theorem, rather modern ...
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277 views

Maximum length of sequence of non-coprimes of $N$ - least upper bound for Jacobsthal's function

I am looking at the length of the longest sequences of adjacent integers that are not coprime to $N$ for very large $N$. Let $F_N$ be the set of integers less than $N$ which are not coprime with $N$: ...
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The divergence of the series of reciprocals of primes (proof check):

I just wanted to check my attempt at a proof for the divergence of: $$\sum_{n=1}^{\infty} \frac{1}{p_n} \tag{ $\star$ }$$ We begin with assuming that $(\star)$ converges. If $(\star)$ ...
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Primes of the form $\frac{n^2-n+4}{2}$ satisfy Hardy-Littlewood analogue?

Let $n,a,b$ be positive integers with $a<b$. Consider primes of the form $f(n)=\dfrac{n^2-n+4}{2}$. Let $C(a,b)$ denote the amount of primes of the form $f(n)$ between (and including) $f(a)$ and ...
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172 views

Fastest Primality test using $N-1$ Factorization?

If $N-1$ could be factored easily with several small prime factors, then what is the fastest way to check $N$ for primality? Updated I'm aware of Pocklington primility test which is not good for ...
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Is the application of $\mu$ on $P_x(s)^k$ analogous to the differentiation $\frac{d^k f(\lambda) }{d\lambda^k}\biggr|_{\lambda=0}$?

Let me start with the following on elementary symmetric polynomials: The elementary symmetric polynomials appear when we expand a linear factorization of a monic polynomial: we have the identity ...
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282 views

Partial summation of a harmonic prime square series (Prime zeta functions)

I am trying to find the following series: $S=\displaystyle\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\dfrac{1}{p_ip_j},A\leq p_1 < p_2 < \dots < p_n \leq B, \lbrace A,B\rbrace \in \mathbb{N}$ ...
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Proof of infinitude of primes whose reversal in base 10 is also prime

Is there any proof of infinitude of http://oeis.org/A007500 primes. If you want to generate them here is trivial and naive python program. ...