Questions tagged [prime-gaps]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1 vote
1 answer
135 views

Infinitely many $k \in \mathbb{N}$ such that the closed interval $[k, k+99]$ contains from $2$ to $23$ prime numbers

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 ...
  • 313
2 votes
0 answers
2k views

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 ...
5 votes
1 answer
226 views

What consequences would follow from the density hypothesis?

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 ...
  • 319
2 votes
1 answer
279 views

Level spacing statistics for primes

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 ...
  • 149
0 votes
2 answers
150 views

How can I convert Meissel's/Lehmer's formula for prime counting to get sum of primes

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 votes
0 answers
95 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 ...
6 votes
0 answers
222 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&...
  • 1,154
2 votes
1 answer
462 views

Is there a Cramer's conjecture for Sophie Germain primes?

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 ...
  • 13.1k
9 votes
1 answer
353 views

The difference between consecutive primes in arithmetic progressions

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 \...
  • 10.7k
5 votes
2 answers
818 views

A question regarding Cramér's proof on prime gaps under the Riemann Hypothesis

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 $C$ ...
  • 898
1 vote
0 answers
220 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} ...
  • 11
0 votes
0 answers
194 views

A question on (trigonometric) prime counting function and twin prime counting function

(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)=\...
  • 620
10 votes
0 answers
266 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 ...
3 votes
0 answers
255 views

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 ...
20 votes
1 answer
558 views

Distinct exponents in the factorization of the factorial, a problem of Erdős

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{...
  • 203
1 vote
0 answers
85 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 ...
0 votes
0 answers
129 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}^...
  • 13.1k
5 votes
0 answers
268 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 ...
2 votes
0 answers
138 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&...
1 vote
1 answer
103 views

Succinct polynomial sized representation of balanced bipartite graphs whose perfect matching count is a primorial

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 ...
  • 13.1k
8 votes
1 answer
386 views

Conjecture about the density of primes

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 votes
0 answers
257 views

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.
  • 181
-4 votes
1 answer
198 views

A generalization Bertrand's postulate [closed]

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 vote
0 answers
74 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 ...
19 votes
1 answer
1k views

Possible contemporary improvement to bounded gaps between primes?

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 ...
  • 31.8k
1 vote
0 answers
176 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。 ...
0 votes
0 answers
49 views

k specific prime factors guess and related prime guess [duplicate]

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 vote
1 answer
221 views

Which gap between primes can be reached under $EH[0.8]$?

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 votes
1 answer
135 views

Upper bound for the number of $k$-central numbers in a prime gap

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 vote
1 answer
149 views

Comparing densities of different gapped primes (twin, cousin, sexy...) [closed]

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 votes
1 answer
122 views

Search for gaps between primes where each composite is divisible by increasing integers (2, 3, 4, ...)

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 votes
0 answers
99 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 $\,...
1 vote
0 answers
151 views

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 $[2n+1]^2$ ...
0 votes
1 answer
215 views

Comparing sets of twin primes with other sets. Why is there a max and min value?

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 votes
0 answers
104 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 ...
6 votes
2 answers
414 views

A generalization of strong primes

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 votes
0 answers
334 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 ...
-3 votes
1 answer
204 views

L. Gegenbauer's proof of Infinitude of Primes [closed]

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 votes
1 answer
751 views

How to explain this prime gap bias around last digits?

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 votes
1 answer
953 views

Some interesting experimental results about the distribution of primes

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 answer
155 views

Numerical estimates for a function relating to twin primes :

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 ...
user avatar
1 vote
2 answers
133 views

Is the abscissa of convergence of $s\mapsto\sum_{n>0}(ng_{n}/2)^{-s}$ known?

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 votes
1 answer
230 views

A conjecture about an inequality that involve Ramanujan primes

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 votes
0 answers
102 views

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 ...
-4 votes
3 answers
621 views

Remarkable articles about the distribution of prime numbers that were written by contemporary physicists [closed]

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 votes
1 answer
291 views

Is there a clear criterion or rule about when one can use the heuristic given by Cramér random model for prime numbers?

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 votes
1 answer
473 views

A weaker version of the Brocard's Conjecture

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 ...
  • 953
19 votes
2 answers
1k views

Is every prime the largest prime factor in some prime gap?

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 votes
1 answer
203 views

Is this theorem on the abundance of prime patterns/k-tuples known?

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 vote
1 answer
182 views

A question about a sum that involves gaps between twin primes, on assumption of the First Hardy–Littlewood conjecture

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{...