Questions tagged [distribution-of-primes]

Use this tag for questions related to the branch of number theory studying distribution laws of prime numbers among natural numbers.

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Distribution of Digits of Binary Expansion of Primes

In considering the binary expansion of prime numbers, I'm interesting in the skew of digits towards 0 or 1. I searched through other questions and arrived at: Last digits of primes I just want to ...
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Algorithm generating subset of primes, can we classify which of them or estimate how large percent of primes are generated?

Assume I have following algorithm: Two lists of numbers, first starting at 2, second starting empty. We now follow rule: Add a number to first list which makes difference with latest number the ...
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A ratio connected to the distribution of primes

According to the Prime Number Theorem, a number $n$, roughly speaking, has probability of primality $\sigma_n:=1/\ln n$. As every schoolchild learns, one can test the primality of $n$ by looking for ...
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Has a Pattern been explored? The sequence of prime factors for a factorial expression?

I have been studying this sequence and I was wondering if there is any pattern to the sequence? Has this been explored? Is there a way to generate an expression that creates these sequences? Is it/...
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Let $s_n$ denote the sum of the first $n$ primes. Prove that for each $n$ there exists an integer whose square lies between $s_n$ and $s_{n+1}$.

Let $s_n$ denote the sum of the first $n$ primes. Prove that for each $n$ there exists an integer whose square lies between $s_n$ and $s_{n+1}$. I cannot give a proof to this, although I have try on ...
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25 views

On particular sets of primes

Let $f,g:\mathbb{N}\to\mathbb{N}$ are two functions. Let $p$ be a prime such that 1) $\#\left\{p\leq x| f(p) \ \text{is also prime}\right\}=+\infty \ \text{as}\ x\to\infty$ 2) $\#\left\{p\leq x| g(...
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How to show that there are infinitely many prime numbers $p$ such that the polynomial f has a zero in $\mathbb Z_p$? [duplicate]

Let $f\in \mathbb Z[X]$ be a polynomial of positive degree.How to show that there are infinitely many prime numbers $p$ such that the polynomial $f$ has a zero in $\mathbb Z/p \mathbb Z$ ? I have no ...
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prime number counting assuming RH: exact or not?

In https://www.quora.com/What-is-the-relationship-between-the-Riemann-Hypothesis-and-prime-numbers and https://en.wikipedia.org/wiki/Prime-counting_function#Exact_form talk of exact formulas for the ...
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Confusion: Proof of Bertrand's Postulate, Primorial function upper bound

My number theory assignment walks me through the proof of Bertrand's postulate. The steps taken are essentially the same as the ones shown here: https://en.wikipedia.org/wiki/Proof_of_Bertrand%...
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80 views

Primes congruent to a mod n.

Is there at least one prime p that is congruent to a mod n, where n can be any positive integer and a can be any non-negative integer less than n?
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66 views

Are there infinitely many primes of the form $k\cdot 2^n+1$ for a fixed $n$ and odd $k$

It is clear from Dirichlet's theorem on arithmetic progressions that for a fixed $n$, there are infinitely many primes of the form $k\cdot 2^n+1$ for a fixed $n$ and $k=1,2,3,..$. However, what if we ...
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Is there an anomaly in the distribution of Mersenne primes?

In a recent press release off the Great Internet Mersenne Prime Search distributed computing project page, it is announced that $$2^{82589933} - 1$$ is the largest known (Mersenne) prime, ...
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estimation for big 'x' in sums over primes

can we state from prime number theorem that $$ \sum_{p\le x}1/p =-\gamma+loglog(x) $$ $$ \sum_{p\le x}p^{m} = li(x^{m+1}) $$ $m$ is a real number $ x\ge -1 $ valid as $ x \sim \infty $
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How this:$\int_{0}^{+\infty} \sin \frac{\pi(1-x)}{2}dx=0 $ with $\pi(x)$ is counting prime function?

let $\pi(x)$ be a the number of prime less than $x$ or prime counting function, I have accrossed in my computation of some integral related to zeta function those two formula : $$\int_{0}^{+\infty} \...
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232 views

Can the exact number of twin primes $\leq n$ be proved using a “twin-prime zeta function”?

Let $\pi(n)$ denote the amount of primes $\leq n$ and let $\pi_2(n)$ the equivalent for twin primes. Properties of $\pi(n)$ can be proved using a well-known formula involving the zeros of the Riemann ...
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107 views

Targeting all prime numbers with a minimal set of sinusoidal functions

Given the series of prime numbers greater than 9, we can organize them in four rows, according to their last digit ($d=1,3,7$ or $9$), and in $k=1,2,3\ldots$ columns corresponding to the $k$-multiple ...
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108 views

Optimal sinusoidal Erathostenes' Sieve

NOTE: I have simplified this post here. Please, consider reading that post instead of this one. Thanks. Given the series of prime numbers greater than 9, we can organize them in four rows, according ...
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61 views

What fraction of primes are $3 (mod 4)$?

As $n\rightarrow\infty$, we know that the fraction of primes that are $2 \hspace{1mm}(mod\hspace{1mm}4)$ tends to zero as two is the only prime with that property. While I am aware that there are ...
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Is there public access to a paper, in which the first $k$-tuple conjecture was proposed? [closed]

How did Hardy and Littlewood derive this conjecture and what needs to be done to prove it?
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The definition of a prime constellation

On Mathworld it is first stated that a prime constellation is a sequence of $k$ prime numbers, for which the gap between the last and the first minimizes. But later they show a table with prime ...
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Primes of form $6n-1$

we know that the probability that a given $n\in\mathbb{N}$ is a prime is $\frac{1}{\log n}$ and all primes except 2 and 3 are of form $6n\mp 1$. We can deduce that the probability that $6n-1$ is $\...
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Generalization of Opperman's Conjecture

Does this conjecture have a name? What about a counterexample?: $$ \forall n,k \in \mathbb{N}, k \gt 1, \exists d \in (kn-n,kn] \text{ s.t. } d \perp n! $$ An equivalent statement is this: Take a ...
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what does the Prime Number Theorem actually say?

I've been reading lots and lots of resources, texts not just blogs and wikis, about the Prime Number Theorem (PNT) and the prime counting approximations. Some sources suggest that the PNT simply ...
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How can I calculate x/log(x) = y for a given y?

I try to test a random prime number generator with a chi-square-test. The prime number generator should generate some prime numbers randomly between [x,y]. But I guess, that the distance between two ...
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70 views

Prove of divergence of reciprocals of primes without infinitude of primes.

I read in a text that Euler once proved the infinitude of primes by proving the divergence of their reciprocals, which seems to me a highly pleasing. However, I am only familiar with one proof of the ...
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What is the difference between Modular Arithmetic and Modulo Operation ?

Trying to rub the dust from my math knowledge, I am trying to understand, or perhaps recall, the Modular Arithmetic and Modulo Operation but I kinda can't get what the differences are between these ...
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4answers
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Twin primes of the form $n^2+1$ and $N^2+3$?

Assume that there are infinity many primes of the form $n^2+1$ and there are infinity many primes of the form $N^2+3$ , Then could we show that there are infinity primes of the form $n^2+1$ and $...
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Generalized Dirichlet theorem for primes in arithmetic sequences

Dirichlet's theorem for arithmetic sequences of primes states that if $a$ and $b$ are coprime, then there exists an infinite number of prime numbers of the form $a+kb$. My question is can we conclude,...
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172 views

Limit involving primes numbers

For each integer $n$ denote by $p_{n}$ the largest prime less or equal than $n$. So, $p_{2}=2$, $p_{3}=3$, $p_{4}=3$, $p_{5}=5$, $p_{6}=5$ and so on. Then, there exists the limit $\lim_{n}\frac{p_{...
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The Basel Problem, Cannonball Problem & 24-ness?

Setting the finite sum of squares equal to a square number yields only one non-trivial solution, when $n= 24$; the sum becomes $4900$, which is $70^2$ $$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$ ...
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109 views

infinitude of primes of special form

Any number prime can only be written as $10k + x$ for $x=1,3,7,9$ since others cannot be primes. Is there a way to prove that there are infinitely many primes of the form $10k+3$ or $10k+7$?
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How to explain twice as much primes like $(2n-1)^2-2$ than like $(2n-1)^2+2$?

Are there explanations why there should be about twice as much primes on the form $(2n-1)^2-2$ than on the form $(2n-1)^2+2$? ...
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Polynomials that often gives primes

Consider $P^f_n=|\{m<n|f(m)\in\mathbb P\}|$, where $f$ is a strictly increasing function $\mathbb N^+\to \mathbb N^+$. Below a diagram of $P^f_n$ for different functions $f$: (Are there ...
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Is the Opposite of the Open Closed in Topology? - On

Following is the topological proof of "infinitude of primes" If you see above proof, it first defines its own toplogy and comments " This topology has two notable properties... 1. the complement ...
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Is the product of primes less than $3\log_2{n}$ always at least $n$?

Consider the product of all primes less than $3 \log_2{n}$. Is it true that this product is always at least $n$ for all positive integers $n$? In general, what is the smallest $x_n$ so that the ...
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How often is the number between two twin primes divided by 6 a prime?

This question has been edited thanks to the feedback by one user: 12 is in between 11 and 13, and 12/6 = 2 which is prime. So if we take 29 and 31, 30 is in between, and 30/6=5 which is prime In ...
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Bounding the number of primes

I have to prove that $$\pi(x)>c\log \log x$$ for some absolute constant $c$. So far I have proven that the nth prime $p_n$ will always be less than $2^{2^{n}}+1$, which I feel maybe helpful.
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Divergence of reciprocal of primes, Euler [duplicate]

On Wikipedia at link currently is: \begin{align} \ln \left( \sum_{n=1}^\infty \frac{1}{n}\right) & {} = \ln\left( \prod_p \frac{1}{1-p^{-1}}\right) = -\sum_p \ln \left( 1-\frac{1}{p}\right) \\ ...
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One particular limit $\lim_{k\to\infty} \frac{k}{p} = \infty$?

Let's take a value of $2^k$ and $p\#$, such that $2^k < p\#$ (but for the lowest possible $p$). As $p\#$ I understand primorial (product of the first $p$ primes). We have: ...
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Prime numbers in factorizations of natural numbers

I would like to know if the following affirmation is true or not: If M is a set of consecutive natural numbers there is a prime number in the factorization of one of M's elements that doesn't divide ...
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Ramanujan Proof of Bertrand Postulate, first step

Here is a link to his proof: http://www.zyymat.com/ramanujans-proof-of-bertrands-postulate.html. I understand the first two "steps", that $$\vartheta(x)=\log2+\log3+\ldots+\log p$$ where $p$ is the ...
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Is this a regularity in primes?

For any prime $p$ subtract $24$ continuously. The last value before $0$ will always be one of these $8$ primes: $\{ 1, 5, 7, 11, 13, 17, 19, 23 \}$. Prime Distribution Across Lengths of 24 Primes in ...
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Prime numbers the rank of which is also a prime.

$127$ has an interesting property: It is the $31$st prime number and its rank ($31$) is also a prime. $31$ is the $11$th prime so its rank is also a prime. $11$ is also a prime number with a rank ($5$)...
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What is really being claimed in this article about a prime number pattern?

I just read this "news" report about Stanford professors who discovered a pattern in prime numbers: https://www.nature.com/news/peculiar-pattern-found-in-random-prime-numbers-1.19550 According to this ...
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59 views

Let $P(k)$ be the $k$-th first prime. Is it true that $P(k+1)\leq P(1)\cdot P(2) \cdots P(k)+1$ [closed]

Let $P(k)$ be the $k$-th first prime. Is it true that $P(k+1)\leq P(1)\cdot P(2)\cdot \dots \cdot P(k)+1$?
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Distribution of divisions of a circle

When viewing distributions divisions of a circle, an interesting behavior is displayed. Take n nested/circumscribed circles. Divide each circle in to n parts and plot a point for each division. ...
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178 views

Prime Numbers - What is the explanation behind this pattern in visualization?

I was playing around with ways to visualize prime numbers as the products of smaller primes. Since a prime is always odd, it can be represented as the product of a set of prime numbers plus one. ...
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82 views

Is the twin of a known prime more likely to be prime?

If I have a large known prime, is the number 2 greater than that or 2 less than that more likely than chance to be prime? It seems that since finding new large primes is something of a newsworthy ...
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237 views

Prime Counting Function from the Sieve of Eratosthenes

This may seem a very simple question, but I did not find any answer for it on the Internet. It is known that the Sieve of Erastothenes can be analytically stated as: $$\pi(x)-\pi(x^{\frac{1}{2}})+1=\...
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1answer
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Prime test by non-polynomial congruence?

I was reading "A Synopsis of Elementary Results in Pure and Applied Mathematics" (available online (http://heybryan.org/docs/A_Synopsis_of_Elementary_Results_in_Pure.pdf)) when I saw the following ...