# Questions tagged [prime-gaps]

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The prime-gaps tag has no usage guidance.

61
questions

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1
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135
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Let $k \in \mathbb{Z}^+$.
Is it possible to prove that, for some given
$m \in \{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23\}$,
there are only finitely many $k$ such that the closed ...

2
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0
answers

2k
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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 ...

5
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1
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226
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Let $N(\sigma,T)$ denote the number of zeros $\rho=\beta+\gamma i$ of the Riemann zeta function satisfying $\beta\ge \sigma$ and $0<\gamma\le T$, counted with multiplicity. Then the "Density ...

2
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1
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279
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In the preprint "Level Spacing Statistics for Primes", we have found some patterns of prime spacings, which may provide new insights on the distribution of primes:
We would like to know ...

0
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2
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150
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Legendre's formula can be very easily be generalised as mentioned here (visible after login) which is like this
${\pi}(v,p)={\pi}(v,p-1)-1.[{\pi}(v/p,p-1)-{\pi}(p-1,p-1)]$
${ \big\downarrow}$
$S(v,p)=...

0
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0
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95
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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 ...

6
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0
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222
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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&...

2
votes

1
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462
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A prime $q$ such that $2q+1$ is also a prime is a Sophie Germain prime.
Cramer's conjecture tells gap between consecutive primes is bound by $O(\log^2p)$.
Is there a similar conjecture for Sophie ...

9
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1
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353
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Let $\pi(x)=\sum_{p\leq x}$ denote the prime counting function. A well known result of Baker, Harman, and Pintz on prime gaps states that for $x\geq y\geq x^{0.525}$ we have that
$$\pi(x+y)-\pi(x)\gg \...

5
votes

2
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818
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Let $p_n$ be the $n$th prime. Assuming the Riemann hypothesis, Harald Cramér proves that $p_n-p_{n-1}\le C(\sqrt p_n \log p_n)$ for sufficiently large $n$. Is there a value known for the constant $C$ ...

1
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0
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220
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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} ...

0
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0
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194
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(I myself don't think the following question is suitable for this forum but as it contains something related to twin primes, I asked here. Please help accordingly)
Consider the following sum:
$$S(t)=\...

10
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0
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266
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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 ...

3
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0
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255
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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 ...

20
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1
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558
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In the 1982 paper below, Paul Erdős proved that if $h(n)$ is the number of distinct exponents in the prime factorization of $n!$ then $$c_1\Big(\frac{n}{\log n}\Big)^{1/2} < h(n) < c_2\Big(\frac{...

1
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0
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85
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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 ...

0
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0
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129
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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}^...

5
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0
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268
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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 ...

2
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0
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138
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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&...

1
vote

1
answer

103
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Is there a $P$ time definable sequence of succinct polynomial sized representation of balanced bipartite graphs whose number of perfect matchings is a primorial?
For factorial a complete bipartite ...

8
votes

1
answer

386
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Conjecture
For any sufficiently large integer $kn$ , the sequence representing
the number of primes in each block obtained by splitting $kn$ into $k$
equal blocks, is a strictly decreasing sequence, ...

6
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0
answers

257
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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.

-4
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1
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198
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Let $n, k$ are integers number such that $1<n \le k$, does always exist a prime number between $kn$ and $k(n+1)$?
When $n=1, k>1$ always exist a prime number between $k$ and $2k$ the question ...

1
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0
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74
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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 ...

19
votes

1
answer

1k
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In his summary of his book Bounded gaps between primes: the epic breakthroughs of the early 21st century, Kevin Broughan writes
Which brings me to my final remark: where to next in the bounded gaps ...

1
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0
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176
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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。
...

0
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0
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49
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there is no more than one group
of continuous composite sequence
of length k composed of only k different specific prime factors.
for example 2 3 5[8 9 10]just only one group. I have prove that k ...

1
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1
answer

221
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This question is a follow-up to Would Elliott-Halberstam conjecture follow from GRH?
Assuming any $\theta<1-\Lambda$ where $\Lambda$ is the de Bruijn-Newman constant is an exponent of distribution ...

0
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1
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135
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Let $I_{n}:=]p_{n},p_{n+1}[$ be the open interval between the $n$-th and $(n+1)$-th prime. Under Goldbach's conjecture, denote by $r_{0}(m)$ the smallest positive integer $r$ such that both $m-r$ and $...

1
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1
answer

149
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In this experiment, I have checked how many times different gapped primes occur out of the first 10000, 100000, 1000000 first primes.
Please view the following as ($X$:$Y$) where $X$ represents the ...

0
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1
answer

122
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Almost every text of number theory contains in its first chapters something similar to the following:
For any integer n, the factorial n! is the product of all positive
integers up to and including n....

2
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0
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99
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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 $\,...

1
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0
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151
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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 $[2n+1]^2$ ...

0
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1
answer

215
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I have taken 2 sets: The first is a consecutive list of the first prime of twin pairs. The second is a consecutive list of numbers as follow 1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5 ....
I have then compared ...

2
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0
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104
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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 ...

6
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2
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414
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In this post we denote the sequence of prime numbers as $p_k$ for integers $k\geq 1$. I don't know if the following definition is in the literature.
Definition. We define the $\theta$-strong primes, ...

5
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0
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334
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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 ...

-3
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1
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204
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I was going through the paper 'Euclid’S theorem on the infinitude of primes: A historical survey of its proofs' by Romeo Mestrovic where he mentioned that
L. Gegenbauer proved Infinitude of Primes by ...

6
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1
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751
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My question is related to this article by Oliver and Soundararajan (article about a bias in the distribution of the last digits of consecutive prime numbers).
After trying some python experimental ...

7
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1
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953
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Let's consider the following metric of the gap between consecutive primes
$$m(k)=\frac {p_k^2-p_{k-1}^2} {24}\;\;\;\;\;(k\ge4)$$
Now, let's define the function
$\delta(k)=m(k)\;\;\;\;$ if $\,m(k)\,$ ...

2
votes

1
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155
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Consider the following function :
$$F(s)= \sum_{\text{$p,\ p+2$ are primes}} \left({\frac{1}{p^s}}+{\frac{1}{(p+2)^s}}\right).$$
Brun's theorem tells us that $F(1)$ is finite.
We are looking for ...

1
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2
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133
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The famous Polignac conjecture posits that the $n$-th prime gap $g_{n}:=p_{n+1}-p_{n}$ attains all even positive integral values infinitely many times, which implies $\displaystyle{\zeta_{Pol}:=s\...

-1
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1
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230
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In this post we denote for integers $n\geq 1$ the $n$-th Ramanujan prime as $R_n$ (thus the sequence A104272 from the On-Line Encyclopedia of Integer Sequences), I add a conjecture that I think can be ...

0
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0
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102
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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 ...

-4
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3
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621
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I would like to ask about if you know papers containing remarkable achievements that were written by contemporary physicists concerning the distribution of prime numbers (or closely related, maybe the ...

3
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1
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291
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I don't know if I have misunderstood the information from the section about the heuristic related to Cramér's conjecture and the known facts about Maier's theorem, from the Wikipedia Cramér's ...

4
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1
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473
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Brocard's conjecture states that: If $p_{k}$ and $p_{k+1}$ are consecutive prime numbers greater than $2$, then between $p_{k}²$ and $p_{k+1}²$ there are at least four prime numbers.
I know that is ...

19
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2
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1k
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Definition: In the gap between any two consecutive odd primes we have one or more composite numbers. One of these composite number will have a prime factor which is greater than that of any other ...

0
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1
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203
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I am looking for references regarding the following statement.
For any two natural numbers x and y there must be a prime k-tuple (a, b, ...) corresponding to x consecutive primes (n+a, n+b, ...) for ...

1
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1
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182
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I wondered, inspired in a result mentioned from [1] (page 45), what should be the asymptotic behaviour of the sequence on assumption of the First Hardy–Littlewood conjecture
$$\sum_{\substack{\text{...