# Questions tagged [prime-gaps]

The prime-gaps tag has no usage guidance.

70
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### Prime gaps that are "relatively" bigger than all later prime gaps: Is this in OEIS?

This OEIS entry is about
Primes (lower end) with record gaps to the next consecutive prime: primes p(k) where p(k+1) - p(k) exceeds p(j+1) - p(j) for all j < k.
I'm wondering about a different ...

3
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0
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### 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 ...

1
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1
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### On the number of primes between prime $p_n$ and $p_{n}^2$

does anyone knows if there are studies on the number of primes between prime $p_n$ and $p_{n}^2$, where $p_n$ is the $n$-th prime?
I am studying it through the following formula:
\begin{align}
\pi(p^...

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1
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### Prime gap conjecture $ \pi_{2a}(n+(6a+4)^3)+(6a+4)^3 > \pi_{4a}(n)$ counterexamples?

Consider prime constellations $p,p+2s$ where both $p,p+2s$ are prime.
For instance for $s=1$ we get the twin primes.
We define the counting function $\pi_{2s}(n)$ to count the number of such pairs $p,...

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1
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278
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### The lower bound for prime gaps

Let $p_n$ denote the $n$-th consecutive prime number and $g_n=p_{n+1}-p_n$ a prime gap. There are many results about the upper bound for $g_n$. Some of them still has astatus of conjecture, such as ...

3
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1
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335
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### What heuristic arguments support Montgomery's conjecture for primes in short intervals?

I have a question regarding a conjecture due to H. L. Montgomery on the number of primes in short intervals. The conjecture apparently arises from probabilistic reasoning upon assuming the Riemann ...

7
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### Some conjectures about prime gaps

I checked some relations between primes, here $1<n<10^5$ and $p_n$ is the $n$th prime.
$a) p_n^{1/3} - p_{n-1}^{1/3}<1/2$
$b) p_n^{1/n} - p_{n-1}^{1/n}<1/n $
$c) (\log p_n)^{1/2} - (\...

3
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0
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### 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 $[x,x+\log^2(x)]$ ...

9
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### Primitive sequences with elements in every interval $[x, x + \sqrt x)$

We believe there is always a prime in the interval $[x,x+\sqrt{x})$, for $x$ sufficiently large, but proving this is inaccessible, even under RH.
What if we just wanted a sequence of integers free of ...

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1
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### 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 ...

1
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0
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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 ...

7
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1
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### 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 ...

2
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1
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### 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 ...

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2
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### 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)=...

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### 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 ...

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### 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&...

2
votes

1
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502
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### 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 ...

9
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1
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378
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### 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 \...

5
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2
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### 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 $C$ ...

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### 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 ...

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0
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235
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### 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} ...

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### 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
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0
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277
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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 ...

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### 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{...

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0
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### 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 ...

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0
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### 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}^...

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### 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
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0
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### 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
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1
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### 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 ...

8
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1
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### 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
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0
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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.

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### 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 ...

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0
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### 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 ...

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1
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### 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 ...

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0
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### 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。
...

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### 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
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1
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### 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
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1
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### 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
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1
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### 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 ...

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1
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### 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....

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0
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### 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 $\,...

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

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1
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### 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 ...

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0
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### 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 ...

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2
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### 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, ...

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### 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 ...

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### 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
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1
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### 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 ...

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1
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### 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 $\,m(k)\,$ ...

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1
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### 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 ...