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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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$H(x)$ approximates $\pi(x)$ pretty well. But what are the drawbacks, when compared with Riemann's $R(x)$?

I'm aware of the Gram series which is equivalent to $R(x)$ (Riemann prime counting function): $$ R(x)=\sum_{n=1}^\infty \frac{\mu(n)}{n}li(x^{1/n}). $$ Over the interval $x=2$ to $x=10^4$ the average ...
zeta space's user avatar
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Primes of the form 2...21

I was wondering what properties could have these numbers: $21, 221, 2221, 22221, ...$ At glance I thought this set would have infinitely many primes. Immediately I went to Python and I realized that ...
Francisco Javier Maciel Hennin's user avatar
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Why number 6 is the most frequent gap when subtracting all consecutive primes(the smaller from the larger)?

Using JavaScript, i felt like collecting all the distances between primes and see what pattern they may have. here is what i got: i generated all primes up to a 1000000, and made an object that counts ...
ZAK's user avatar
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Density of Primes in different sets

I am examining the density of Primes in other sets than the naturals. E.g. we want to have the density of Mersenne Primes. From the prime number theorem I know that in the naturals we have $$ \frac{\...
Lereu's user avatar
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Is it true that for any positive integer $n$, there exists an integer $x$ where there are at least $n$ primes between $x^2$ and $(x+1)^2$

Am I correct that this follows directly from two observations: (1) The sum of the reciprocals of primes diverges. (2) The sum of the reciprocals of squares converges Here's my thinking: If there ...
Larry Freeman's user avatar
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3 answers
55 views

Examples of integer sequences that have a distribution approx $1/\log(n)$, like the primes do?

It is well known that the primes are distributed such that they occur with an approximate "likelihood" of $1/\log(n)$ around the integer $n$ - or more precisely, the number of primes up to $...
Penelope's user avatar
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Strange claim about the number of primes between two given numbers.

I recently saw a meme posted to the Facebook group "Mathematical Mathematics Memes." The meme listed some results about the number of primes between two given numbers. The last result listed ...
VShaw's user avatar
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Approximating the logarithms of primes elegantly

What's the "most efficient" way to encode that $\ln(2):\ln(3):\ln(5):\ln(7) \approx 171:271:397:480$ using $3$ approximations? For example: $((\frac{3^3}{5^2})^3)^3 \approx 2$ uses $3+2+3+3 ...
Chinmay The Math Guy's user avatar
3 votes
1 answer
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What is the order of $\mathrm{Z}(x)-\pi(x)?$

Consider the offset logarithmic integral which approximates the number of primes up to a given $x$ quite well $$ \mathrm{Li}(x)=\int_2^x\frac{1}{\log t}~dt$$ And consider the alternating series $$\...
zeta space's user avatar
3 votes
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Why is the Riemann explicit formula for primes (via the Riemann Hypothesis) any better than existing formulae?

Many people, such as here, don't consider Willan's formula for primes, and other such formulas given here, as meaningful formulae for computing primes. My understanding of the main criticisms are that ...
Tanishq Kumar's user avatar
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Why do number theorists care so much about how well $\text{Li}(x)$ approximates $\pi(x)$ if it's not our best approximation?

An alleged primary motivator for the RH is so that we can bound the error term $|\text{Li}(x) - \pi(x)|$ by a factor of $O(\sqrt{x}\log x)$. However, I also learned about Riemann's explicit formula $R(...
Tanishq Kumar's user avatar
2 votes
2 answers
155 views

Asymptotics for $g(n) = \sum_{k = 1}^{n - 1} {\frac{{\log (1 + p_k)}}{p_k}}$?

Let $p_k$ be the $k$th prime. Now define $g(n)$ as $$g(n) = \sum_{k = 1}^{n - 1} {\frac{{\log (1 + p_k)}}{p_k}}$$ What are the asymptotics for this $g(n)$ ? The related sum $$ \sum_{k = 1}^{n - 1} {\...
mick's user avatar
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Stronger result than bertrand's postulate

It is well known that there is a prime number between $n$ and $2n$ for all $n$. I decided to go deeper: is there a lower bound on the number of primes between $n$ and $2n$ for "large enough" ...
Joseph Martin's user avatar
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The Density of Primes

The below discussion comes from HM Edwards book on the Riemann Zeta Function. (2′)$$\sum_{p < x}1/p∼\log(\log𝑥)\,\,(𝑥→∞)$$, (...) Now $$\log(\log𝑥)= \ \int_{1}^{\log(x)} \frac{du}{u} \ = \int_{e}...
L. Tim's user avatar
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Primitive divisors of Fibonacci numbers

The well known Fibonacci sequence $F_{0} = 0, F_{1} = 1$ and, by recurrence law, $F_{n+1}:=F_{n} +F_{n-1}$ for all $n\geq 1$, has the following property (proved by Carmichael in 1913): With the ...
user123043's user avatar
6 votes
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113 views

Are there infinitely many primes less than $q^{1+\epsilon}$ equivalent to $1$ mod $q$?

Fix $\epsilon>0$. As $q$ becomes large, is it true that the number of primes less than $q^{1+\epsilon}$ congruent to $1$ modulo $q$ will tend to infinity? A conjecture of Montgomery says that the ...
Milo Moses's user avatar
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Is there a close approximation for $\prod_{i=1}^n (1-1/p_i)$?

Consider the primes $p_i$ starting at $p_1 = 2$. Is there a close approximation for $$\prod_{i=1}^n (1-1/p_i)\;?$$ Plotting the values you get the following.
Simd's user avatar
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Given $n = ab$ with $a,b$ unknown what are some good choices for $a_0, d$ such that $a,b$ are both in the AP $a_0 + kd$

Problem: Given $n = ab > 0$ with $a,b \in \mathbb{Z}$ unknown, what are some good choices for $a_0, d$ such that $a,b$ are both in the AP $a_0 + kd$? We are not making any assumptions about $a,b$ ...
vvg's user avatar
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Obtaining strings of congruent primes (Shiu's Theorem)

While stumbling upon this question, I came across this interesting Theorem of Daniel Shiu: Theorem (Shiu, 2000): Let $p_n$ denote the $n$-th prime. $\forall k \in \mathbb{N}$ and $a, q \in \mathbb{N}, ...
vvg's user avatar
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Collinear primes $(n,p_n)$

This is about prime numbers such that at least three points $(i,p_i)$, $(j,p_j)$ and $(k,p_k)$ are on the same straight line. Conjectures: For any pair $(i,p_i),\, i>1$, there are two different ...
Lehs's user avatar
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Is there really no way to generate infinitely many primes?

Is there really no way to generate infinitely many primes? A previous answer for someone asking about the Infinite generation of primes, says: There is no exact way to generate primes continuously. ...
Malady's user avatar
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A sequence is defined by the nth term: $n^3 -21n^2 +99n +121$. What primes does the sequence contain if continued to infinity? [closed]

A sequence is defined by the nth term: $$n^3 -21n^2 +99n +121$$ What primes does the sequence contain if continued to infinity? (Question given by maths teacher in stretch and challenge workshop)
Mathematician's user avatar
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1 answer
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angular distribution of primes in a number field

For my master's thesis, I need to work on a problem related to the "angular" distribution of primes in a Number field. For the sake of simplicity, let's take a real quadratic number field K. ...
Yashi Jain's user avatar
6 votes
1 answer
337 views

A prime generating algorithm

I was trying to explain the famous proof of infinitude of primes to a young one, and I tried to explicitly show some examples. So, I said something like Let the only primes be $2,3,5$. Then $$N=2\...
Sayan Dutta's user avatar
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Estimation of sum of prime power product

In the book 'Probabilistic Number Theory I Mean- Value Theorems' by P.D.T.A. Elliott the writer mentioned the following fact without proof. Fact: $$ \sum_{\substack{p\neq q\\ p^k q^l\leq x}} \!\! p^k ...
Math is beautiful's user avatar
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2 answers
65 views

Can you prove my Prime assumption wrong?

Right, so I was playing about with prime numbers and came across something very odd and interesting. Check this out. IF $\frac{3x+1}{2^n} \in 2\mathbb{N}-1$ WHERE $n \in \mathbb{N}, x \in 2\mathbb{N}-...
UndercoverCoder's user avatar
1 vote
1 answer
94 views

Primality Formula Conjecture

Primality Formula Conjecture To test any $(6x-1)$ numbers for primality. $$ 4^{3x-1} \bmod (6x-1) $$ If that is equal to 1 then $(6x-1)$ is prime. To test any $(6x+...
jaredjbarnes's user avatar
5 votes
1 answer
319 views

Does the truth of one imply the other? A simple Collatz generalization in terms of primes.

Let $f_i:\mathbb{N} \to\mathbb{N}$. The Collatz function states that the following iterated map will eventually equal to 1: $$f_0(n) = \begin{cases} n/2, & \text{if}\ 2\mid n\\ 3n+1, & \text{...
Math777's user avatar
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Variations on $\pi(2n)-\pi(n)$

I have taking a glance to the sequence $a_n=\pi(2n)-\pi(n)$ for $n\in\Bbb N$, where $\pi$ is the prime counting function ($\Bbb N$ does not include $0$ in this post). You can see some terms here: ...
ajotatxe's user avatar
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3 votes
1 answer
231 views

Distribution of prime gaps - is it an unsolved problem?

Numerical experiments show the distribution of prime gaps conforms to some quite firm constraints. The following plot visualises these constraints - it shows the log of the count of prime gaps against ...
Penelope's user avatar
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1 vote
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Is there a bound of $\Delta(x)$ related to Riemann prime-counting funtion?

$$\Delta (x)=\left(\pi (x)-\operatorname {R} (x)+{\frac {1}{\ln x}}-{\frac {1}{\pi }}\arctan {\frac {\pi }{\ln x}}\right){\frac {\ln x}{\sqrt {x}}},$$ where $\operatorname{R}(x)$ is Riemann R function....
Lapin's user avatar
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1 answer
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Stuck on Derivation of Chebyshev's Estimates for $\pi(n)$

I am trying to develop a simple derivation of Chebyshev's estimates for the prime counting function $\pi(x)$. Stopple's "A Primer of Analytic Number Theory" gives a good start: By ...
Penelope's user avatar
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"Stronger" form of Dirichlet's theorem on primes in arithmetic progressions

Let $a$ and $q$ be two relatively prime positive integers. If I know that $$\lim_{s\rightarrow 1^{+}}\sum_{p\equiv a \bmod q}\frac{1}{p^s}=+\infty,$$ then it is clear that it tells me that there are ...
Sqrt's user avatar
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1 vote
1 answer
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Asymptotic of prime number pairs

I am not an expert in number theory, so if someone might know the answer or a lead for the following, it would be greatly appreciated. Let $\mathcal{P}$ be the primes, and let $x$ stand for a natural ...
Mahir Lokvancic's user avatar
1 vote
0 answers
167 views

Why is the Theorem of Sylvester and Schur a generalization of Bertrand's postulate?

The Theorem of Sylvester and Schur states that for some n>k, one of n+1,...,n+k-1, has a prime divisor exceeding k. While Bertrand's postulate says that for all positive integers n, there exists a ...
GraphMathTutor's user avatar
2 votes
1 answer
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Does there exist a a general counting function related to the prime counting function?

Does there exist a a general counting function related to the prime counting function? Say for example I wanted all the positive integer multiples of three less than or equal to N, is there a ...
user avatar
0 votes
1 answer
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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 ...
عبد الرحمن رمزي محمود's user avatar
2 votes
1 answer
72 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 ...
Trevor's user avatar
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0 votes
1 answer
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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 $a$ ...
عبد الرحمن رمزي محمود's user avatar
3 votes
1 answer
88 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 ...
user avatar
4 votes
0 answers
80 views

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, ...
Joseph O'Rourke's user avatar
1 vote
1 answer
63 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\...
Milo Moses's user avatar
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1 vote
1 answer
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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 $O(\frac{x}{\log^2(x)})$. ...
UnusualMathem's user avatar
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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 ...
عبد الرحمن رمزي محمود's user avatar
1 vote
1 answer
88 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 ...
bozcan's user avatar
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1 vote
2 answers
39 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 ...
Noomkwah's user avatar
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1 vote
0 answers
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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 ...
Tim Hargreaves's user avatar
0 votes
2 answers
44 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 $...
OrenIshShalom's user avatar
1 vote
0 answers
49 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, $$ ...
zeta space's user avatar
1 vote
1 answer
1k 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 ...
Legorooj's user avatar
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