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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a question about the distribution of primes

Definition : we shall call $p$ a special prime $p_n$, if there is at least one prime of the form $2kp+1$, where $ 1 \leq k \leq n$ . it is obvious that we can call any Sophie Germain prime, a special ...
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48 views

(soft) Can an efficient closed-form expression for $P_n$ be found? [closed]

I just read several old threads on here with people asking about formulas for primes, and what the implications of having one would be. As everyone was quick to point out, we already have a bunch, in ...
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44 views

A question about the probability of being a prime?

If we chose a random number $a \leq N$, then, the probability for $a$ to be a prime is $\frac{1}{\log N}$. Now, if there are some primes that do not divide $a$, then what is the probability for $a$ ...
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54 views

is there any possibility to write and calculate this sum in pari gp, which is very related to hardy littlewood first conjecture?

I studied hardy littlewood first conjecture, which predicts the density of primes of special form, so: if I want to know the number of the primes of the form $2kp+1$, where $p$ is prime and $p \leq ...
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Do lucky numbers contain arbitrarily long arithmetic progressions?

The lucky numbers are defined by a sieve, which results in numbers that asymptotically mirror the prime density $\sim n / \log n$: $$ 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, ...
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57 views

Show that if $\lVert a_{pn}\rVert>0$ for primes $p$, then $\lVert a_n \rVert\geq0$

If we define $$\lVert a_n\rVert = \lim_{N\to\infty}\frac{1}{N}\sum_{k=0}^{N-1}a_k$$ To be the mean of a sequence, and we let $a_n$ be a bounded sequence of integers where not only does $\lVert a_n\...
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53 views

A Selberg's sieve application

I have a problem concerning Selberg's sieve: I have to prove that, given $k>1$ a fixed even integer, the number of primes $p\leq x$ such that $1+pk$ is still a prime is $O(\frac{x}{\log^2(x)})$. ...
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39 views

how to predict the density of these prime numbers?

if I wanna know how many prime number $p$ less than $x$, such that : there is at least one prime number of this form$ 2Kp+1$ where $K={1,2,3,4,5,6,7,8,9,10}$. for example: if $x=1000$, then the ...
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1answer
53 views

On Bertrand's Postulate

By Bertrand's postulate, we know that there exists at least one prime number between $n$ and $2n$ for any $n > 1$. In other words, we have $$ \pi(2n) - \pi(n) \geq 1, $$ for any $n > 1$. The ...
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2answers
26 views

Prime reciprocals series proof

I am currently struggling with the proof of the prime reciprocals series divergence. I've already proved that : $$\prod_{k=1}^n \frac{1}{1-1/p_k} \longrightarrow +\infty$$ Let $n$ be an integer. How ...
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Editing digits to find primes

Suppose that I am given a random, large (900+ digits) integer. I would like to edit a few of its digits so that the resulting integer is a prime. I am quite flexible on the definition of 'a few'. I ...
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37 views

$M\in\mathbb{N}$ such that $\pi(n)\leq\frac{Mn}{\log n}$

I know that $\pi(n)$ is approximately $\frac{n}{\log n}$. Is there any constant $M \in \mathbb{N}$ that satisfies: $\forall n \in \mathbb{N}~.~\pi(n) \leq \frac{Mn}{\log n}$ I need the upper bound $...
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46 views

Any deducible pattern in this sequence? Is it related to primes?

Here's a part of a sequence that I want to understand better: $S=\{0,0,1,1,2,2,3,2,4,3,4,3,5,4,5,4,6,5,7,5,8,5,7,5,8,7,...\}.$ The sequence comes from finding distinct values for $f_n(x)$ where, $$ ...
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1answer
73 views

How to calculate how many primes there are of $N$ bits

Is there a way to calculate how many primes are exactly $N$ bits in length, without generating them? I know that you can calculate how many primes are below $N$, but not how/if you can calculate ...
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58 views

How to proof the following property? [duplicate]

Today I was studying a correctness proof of an algorithm, and in one of the proof steps and did not understand what was written. Be $p$ and $n \in N$ ($N$ is the set of naturals numbers), where $p$ ...
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144 views

Small gaps between primes Maynard-Tao sieve why it works

TLDR: How do we know the probability distribution that Maynard obtains in the small gaps between primes problem is $\mathbb{P}(n+h_{i}) \asymp \frac{\log k}{k}$. I'm interested in why the sieve ...
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1answer
60 views

On the sums of primes

It is known that: $$\sum_{k=1}^np_k \sim \frac12n^2\ln{n}$$ The proof of which I cannot find. Here is a related MSE post with an appreciable answer. Now, I wonder how one might go about finding $f(n)...
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49 views

Why this strategy is true…

To find prime number if we want to be sure, why can't we just put odd number of zeros between two ones and get a prime number, and also why this becomes true and how to prove it. Example: 101 10001 ...
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74 views

Density of Numbers with Exactly One Prime Factor of Multiplicity 1

Let $S$ be the set of positive integers $n$ with the property that exactly one prime factor of $n$ has multiplicity $1$ and every other prime factor has multiplicity greater than $1$ (to be clear, $S$ ...
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1answer
45 views

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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1answer
47 views

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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30 views

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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86 views

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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30 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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102 views

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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73 views

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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55 views

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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238 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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73 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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116 views

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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40 views

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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1answer
311 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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1answer
170 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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121 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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96 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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49 views

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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161 views

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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66 views

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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1answer
169 views

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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59 views

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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74 views

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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1answer
103 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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280 views

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
135 views

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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51 views

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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202 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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130 views

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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2answers
121 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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81 views

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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76 views

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 ...