Questions tagged [prime-gaps]
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26
questions with no upvoted or accepted answers
28
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answers
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Does this infinite primes snake-product converge?
This re-asks a question I posed on MSE:
Q. Does this infinite product converge?
$$
\frac{2}{3}\cdot\frac{7}{5}\cdot\frac{11}{13}\cdot\frac{19}{17}\cdot\frac{23}{29}\cdot\frac{37}{31} \cdot \cdots \...
10
votes
0
answers
322
views
Are there are any attempts utilising sieve theory to attack the general $a p \pm 1$ problem?
It is currently an open question if there are infinitely many primes $p$ such that $2p + 1$ is prime (Sophie Germain primes) or that at least one of $24p \pm 1$ is prime.
Could Zhang's method, or the ...
6
votes
0
answers
422
views
On improvements of the GPY sieve
When $\chi_\mathbb P(n)$ denotes the characteristic function of primes and $\mathcal H=\{h_1,h_2,\dots,h_k\}$ is some admissible $k$-tuple, the GPY sieve can be formulated as follows:
$$
S(x)=\sum_{x&...
6
votes
0
answers
306
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Hoheisel's paper
Does anyone know a digital link to Hoheisel's paper: "Primzahlprobleme in der Analysis"?
It appeared in the 30's, published by the Berlin Academy. There seems to be no digital version.
6
votes
0
answers
132
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Upper bound for number of primes close to the next prime
Let $p_n$ denote the $n$th prime number and let $g_n := p_{n+1} - p_n$ be the $n$th prime gap. I'm looking for a good upper bound for the quantity
$$G(x, y) :=\#\{p_n \leq x : g_n \leq y\} ,$$
holding ...
5
votes
0
answers
314
views
Counting primes, twin primes, cousin primes: unusual approach, connection to some conjectures
I am investigating the following sieve-like algorithm. Let $S_N=\{1,\dots,N\}$. For all primes $p$ with $p_0\leq p \leq M$, we remove from $S_N$ the following elements: all numbers $n\in S_N$ such ...
5
votes
0
answers
338
views
On a conjecture about the arithmetic function that counts the number of twin primes
This is cross-posted from the question that I've asked with same title on Mathematics Stack Exchange two months ago, which has remained unanswered.
Given a positive real number $x$ we will write ...
5
votes
0
answers
281
views
Symmetry of the distribution of prime gaps
Following Positive proportion of logarithmic gaps between consecutive primes let for given $\lambda$, $\alpha$ and for any $x$ all positive the quantities $S^{-}_{\lambda,\alpha}(x):=\#\{p_{n+1}\...
3
votes
0
answers
232
views
The number of continuously increasing primes gaps in the interval $[2,n]$ is less than $\log n$
A prime gap is the difference between two successive prime numbers. The $n$-th prime gap, denoted gn or $g(p_n)$ is the difference between the $(n+1)$-st and the $n$-th prime numbers. Using my ...
3
votes
0
answers
67
views
Divisor of given order in short intervals
Is the following Open question or Conjecture already known, or eventually settled ?
Open question : For sufficiently large $x$ there is at least a positive integer in the interval $[x,x+\log^2(x)]$ ...
3
votes
0
answers
280
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A prime generating algorithm
I posted this question in MSE around a month ago, but didn't receive any suitable answers. So, I decided to give it a try here as well-
I was trying to explain the famous proof of infinitude of primes ...
3
votes
0
answers
125
views
Number of prime differences
Has any progress been made since Chen on bounding
\begin{equation*}
G(n) = \#\{\epsilon N < p_1, p_2 \leq N: n = p_1 - p_2\}
\end{equation*}
from above?
As far as I can tell, the best upper ...
2
votes
0
answers
93
views
Primes as expected values?
This is a follow-up question, which is related to the answer of this quesiton: Is there a connection of prime numbers and extreme value theory?
I will duplicate the answer here, so this question is ...
2
votes
0
answers
142
views
Measuring philoprimality/misoprimality
Given a natural integer $x$, let
$$\alpha(x)=(\log x)^2\sum_{p\in\mathcal P\setminus\{x\}}\frac{1}{(x-p)^2}$$
(with $\mathcal P$ denoting the set of prime-numbers)
measure its "philoprimality&...
2
votes
0
answers
110
views
A conjectured upper bound for the mean value of prime divisors inside prime gaps
In 1969 C.A. Grimm stated this interesting conjecture: the prime gap $\,G_n=\{x\in N:p_n\lt x\lt p_{n+1}\}\,$ contains at least $\,\#G_n=(p_{n+1}-p_n)-1=g_n-1\,$ distinct prime divisors, that is if $\,...
2
votes
0
answers
107
views
Prime gap transform
Let $n$ be a large enough composite integer, and consider an arithmetic function $f$ that maps $n$ to the sum of prime gaps making a closed interval $J_{f}(n)$ containing $n$ whose extremities are ...
1
vote
0
answers
3k
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Contribution of Yitang Zhang latest results if correct to correlation conjecture of H. L. Montgomery?
There are some integrals related to the celebrated pair correlation conjecture of H. L. Montgomery. One of them is the integral introduced by Selberg related to estimating the variance of primes in ...
1
vote
0
answers
237
views
Liu's new sieve weight
Does Liu's sieve weight (in his arXiv paper "On the gap between primes")
$$sieve(n)=(\sum_{\substack{d_i\mid (n-h_i),i=1,\cdots,k\\ (d_1,\cdots,d_k)\in\mathcal{D}}}\lambda_{d_1,\cdots,d_k} ...
1
vote
0
answers
95
views
Sum of reciprocals of maximal prime gaps and primes
Let $G_r =$ http://oeis.org/A005250, and $P_r =$ http://oeis.org/A002386.
$\sum_{n=1}^{\infty}{\frac{1}{G_r}} = c_1$
$\sum_{n=1}^{\infty}{\frac{1}{P_r}} = c_2$
Do the constants c_1 and c_2 exist?
The ...
1
vote
0
answers
87
views
Prime numbers and gaps of multiplications of triangular numbers
Triangular numbers: $T_n = \frac{n(n+1)}{2} = 1,3,6,10,15,21,28...$
From my observations of the first $10000$ primes:
For any prime $P$ greater than $3$:
Observation 1) There will always be at least ...
1
vote
0
answers
183
views
Do $k$ specific prime factors uniquely determine the continuous composite sequence of length $k$?
猜想:不存在两个长度为k且一共含有k个不同素因子的连续合数序列的素因子集合相等。例如
24 25 26 27 (2 3 5 13)
其余长度为4的连续合数序列要么素因子个数大于4 要么素因子集合不等于{2 3 5 13} 这个连续合数猜想的重大特定情况下的猜想。难度可与哥德巴赫猜想匹敌,非天才误入。再比如2 3 5 唯一决定了8 9 10 除此之外再无其他。总之经过计算机验证没有找到反例。
...
1
vote
0
answers
170
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Related to one of the twin prime conjectures (In between squares)
This question is inspired by an answer that I have received for another question [see here]:
One of the twin prime conjectures states that Between the squares of two consecutive odd numbers $[2n+1]^2$ ...
1
vote
0
answers
277
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Prime numbers in this region
Let $q \geq 5$ be a prime number, and consider : $N_q = \displaystyle{\small \prod_{\substack{p \leq q \\ \text{p prime}}} {\normalsize p}}$
Using Chinese remainder theorem we can show that :
$$\#\{(...
0
votes
0
answers
103
views
If we weaken Polignac's conjecture to an existential claim, can it be proved?
Polignac's conjecture (unproved) states that, for any integer $k \geq 1$, there exist infinitely many $p$ such that $p$ and $p+2k$ are both prime. Suppose that we weaken the consequent to require only ...
0
votes
0
answers
134
views
On a deterministic primes search problem
I feel the following problem might be resolved already. But I could not find any related answers.
If $p_1,p_2,\dots,p_t$ are primes where $2\leq t=o(\log n)$ is there a prime within $$\prod_{i=1}^...
0
votes
0
answers
104
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Variants of Nicholson's inequalities for prime numbers, involving the Lambert $W$ function
The purpose of this post is ask about two closely related/inspired conjectures from inequalities due to Nicholson (see [1]) and Visser [2]. If my reasonings are right should be stronger versions of ...