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.
120
questions
0
votes
0
answers
63
views
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 ...
- 141
2
votes
0
answers
58
views
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 ...
3
votes
1
answer
79
views
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
$$\...
- 892
3
votes
0
answers
149
views
Why is the Riemann explicit formula for primes (via the Riemann Hypothesis) any better than existing formulae?
Many people, such as here, don't consider Willan's formula for primes, and other such formulas given here, as meaningful formulae for computing primes. My understanding of the main criticisms are that ...
- 651
10
votes
1
answer
245
views
Why do number theorists care so much about how well $\text{Li}(x)$ approximates $\pi(x)$ if it's not our best approximation?
An alleged primary motivator for the RH is so that we can bound the error term $|\text{Li}(x) - \pi(x)|$ by a factor of $O(\sqrt{x}\log x)$. However, I also learned about Riemann's explicit formula $R(...
- 651
2
votes
2
answers
144
views
Asymptotics for $g(n) = \sum_{k = 1}^{n - 1} {\frac{{\log (1 + p_k)}}{p_k}}$?
Let $p_k$ be the $k$th prime.
Now define $g(n)$ as
$$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} {\...
- 15.3k
8
votes
1
answer
175
views
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" ...
- 139
0
votes
0
answers
29
views
Pattern for the count of numbers whose smallest prime factor is $p $ in a defined range
I was studying numbers whose smallest prime factor is $p$, then it was interesting for me to find how many of them underly between $0$ and a defined range. I made the calculations for primes up to $...
- 100
3
votes
1
answer
170
views
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
$$\log(\log𝑥)= \ \int_{1}^{\log(x)} \frac{du}{u} \ = \int_{e}...
- 231
4
votes
1
answer
157
views
Primitive divisors of Fibonacci numbers
The well known Fibonacci sequence $F_{0} = 0, F_{1} = 1$ and, by recurrence law, $F_{n+1}:=F_{n} +F_{n-1}$ for all $n\geq 1$, has the following property (proved by Carmichael in 1913):
With the ...
- 174
6
votes
0
answers
112
views
Are there infinitely many primes less than $q^{1+\epsilon}$ equivalent to $1$ mod $q$?
Fix $\epsilon>0$. As $q$ becomes large, is it true that the number of primes less than $q^{1+\epsilon}$ congruent to $1$ modulo $q$ will tend to infinity?
A conjecture of Montgomery says that the ...
- 2,507
3
votes
2
answers
130
views
Is there a close approximation for $\prod_{i=1}^n (1-1/p_i)$?
Consider the primes $p_i$ starting at $p_1 = 2$. Is there a close approximation for
$$\prod_{i=1}^n (1-1/p_i)\;?$$
Plotting the values you get the following.
- 291
-1
votes
1
answer
28
views
Given $n = ab$ with $a,b$ unknown what are some good choices for $a_0, d$ such that $a,b$ are both in the AP $a_0 + kd$
Problem: Given $n = ab > 0$ with $a,b \in \mathbb{Z}$ unknown, what are some good choices for $a_0, d$ such that $a,b$ are both in the AP $a_0 + kd$? We are not making any assumptions about $a,b$ ...
- 3,205
1
vote
0
answers
72
views
Obtaining strings of congruent primes (Shiu's Theorem)
While stumbling upon this question, I came across this interesting Theorem of Daniel Shiu:
Theorem (Shiu, 2000): Let $p_n$ denote the $n$-th prime. $\forall k \in \mathbb{N}$ and $a, q \in \mathbb{N}, ...
- 3,205
3
votes
0
answers
62
views
Collinear primes $(n,p_n)$
This is about prime numbers such that at least three points
$(i,p_i)$, $(j,p_j)$ and $(k,p_k)$ are on the same straight line.
Conjectures:
For any pair $(i,p_i),\, i>1$, there are two different ...
- 13.8k
0
votes
1
answer
133
views
Is there really no way to generate infinitely many primes?
Is there really no way to generate infinitely many primes?
A previous answer for someone asking about the Infinite generation of primes, says:
There is no exact way to generate primes continuously.
...
- 207
-1
votes
1
answer
78
views
A sequence is defined by the nth term: $n^3 -21n^2 +99n +121$. What primes does the sequence contain if continued to infinity? [closed]
A sequence is defined by the nth term:
$$n^3 -21n^2 +99n +121$$
What primes does the sequence contain if continued to infinity?
(Question given by maths teacher in stretch and challenge workshop)
1
vote
1
answer
86
views
angular distribution of primes in a number field
For my master's thesis, I need to work on a problem related to the "angular" distribution of primes in a Number field. For the sake of simplicity, let's take a real quadratic number field K. ...
- 29
6
votes
1
answer
335
views
A prime generating algorithm
I was trying to explain the famous proof of infinitude of primes to a young one, and I tried to explicitly show some examples. So, I said something like
Let the only primes be $2,3,5$. Then
$$N=2\...
- 8,485
1
vote
1
answer
77
views
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:
$$
\sum_{\substack{p\neq q\\ p^k q^l\leq x}} \!\! p^k ...
-1
votes
2
answers
65
views
Can you prove my Prime assumption wrong?
Right, so I was playing about with prime numbers and came across something very odd and interesting. Check this out.
IF $\frac{3x+1}{2^n} \in 2\mathbb{N}-1$
WHERE $n \in \mathbb{N}, x \in 2\mathbb{N}-...
- 111
1
vote
1
answer
89
views
Primality Formula Conjecture
Primality Formula Conjecture
To test any $(6x-1)$ numbers for primality.
$$
4^{3x-1} \bmod (6x-1)
$$
If that is equal to 1 then $(6x-1)$ is prime.
To test any $(6x+...
- 31
4
votes
1
answer
312
views
Does the truth of one imply the other? A simple Collatz generalization in terms of primes.
Let $f_i:\mathbb{N} \to\mathbb{N}$. The Collatz function states that the following iterated map will eventually equal to 1:
$$f_0(n) =
\begin{cases}
n/2, & \text{if}\ 2\mid n\\
3n+1, & \text{...
- 682
1
vote
0
answers
54
views
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: ...
- 64.8k
3
votes
1
answer
197
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 ...
- 2,523
1
vote
0
answers
44
views
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....
- 160
0
votes
1
answer
76
views
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 ...
- 2,523
0
votes
0
answers
53
views
"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 ...
- 207
1
vote
1
answer
114
views
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 ...
1
vote
0
answers
147
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 ...
- 472
2
votes
1
answer
66
views
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 ...
user817548
0
votes
1
answer
63
views
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 ...
2
votes
1
answer
64
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 ...
- 5,739
0
votes
1
answer
84
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 $a$ ...
3
votes
1
answer
85
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 ...
user788522
4
votes
0
answers
79
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 $\sim n / \log n$:
$$
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, ...
- 30.3k
1
vote
1
answer
63
views
Show that if $\lVert a_{pn}\rVert>0$ for primes $p$, then $\lVert a_n \rVert\geq0$
If we define
$$\lVert a_n\rVert = \lim_{N\to\infty}\frac{1}{N}\sum_{k=0}^{N-1}a_k$$
To be the mean of a sequence, and we let $a_n$ be a bounded sequence of integers where not only does $\lVert a_n\...
- 2,507
1
vote
1
answer
115
views
A Selberg's sieve application
I have a problem concerning Selberg's sieve: I have to prove that, given $k>1$ a fixed even integer, the number of primes $p\leq x$ such that $1+pk$ is still a prime is $O(\frac{x}{\log^2(x)})$.
...
- 187
0
votes
0
answers
58
views
how to predict the density of these prime numbers?
if I wanna know how many prime number $p$ less than $x$, such that :
there is at least one prime number of this form$ 2Kp+1$ where $K={1,2,3,4,5,6,7,8,9,10}$.
for example:
if $x=1000$, then the ...
1
vote
1
answer
79
views
On Bertrand's Postulate
By Bertrand's postulate, we know that there exists at least one prime number between $n$ and $2n$ for any $n > 1$. In other words, we have
$$
\pi(2n) - \pi(n) \geq 1,
$$
for any $n > 1$.
The ...
- 1,173
1
vote
2
answers
39
views
Prime reciprocals series proof
I am currently struggling with the proof of the prime reciprocals series divergence.
I've already proved that :
$$\prod_{k=1}^n \frac{1}{1-1/p_k} \longrightarrow +\infty$$
Let $n$ be an integer. How ...
- 441
1
vote
0
answers
57
views
Editing digits to find primes
Suppose that I am given a random, large (900+ digits) integer. I would like to edit a few of its digits so that the resulting integer is a prime. I am quite flexible on the definition of 'a few'. I ...
- 309
0
votes
2
answers
44
views
$M\in\mathbb{N}$ such that $\pi(n)\leq\frac{Mn}{\log n}$
I know that $\pi(n)$ is approximately $\frac{n}{\log n}$.
Is there any constant $M \in \mathbb{N}$ that satisfies:
$\forall n \in \mathbb{N}~.~\pi(n) \leq \frac{Mn}{\log n}$
I need the upper bound $...
1
vote
0
answers
48
views
Any deducible pattern in this sequence? Is it related to primes?
Here's a part of a sequence that I want to understand better:
$S=\{0,0,1,1,2,2,3,2,4,3,4,3,5,4,5,4,6,5,7,5,8,5,7,5,8,7,...\}.$
The sequence comes from finding distinct values for $f_n(x)$ where, $$ ...
- 892
1
vote
1
answer
899
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 ...
- 113
0
votes
1
answer
74
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 $p$ ...
3
votes
0
answers
760
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 ...
- 571
2
votes
1
answer
84
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)...
0
votes
0
answers
53
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
...
6
votes
0
answers
141
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, $S$ ...
- 2,687