Questions tagged [prime-gaps]

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Prime arithmetic triples [closed]

I wonder if this statement holds. For any prime $p \ge 5$, there exists an integer $d$ such that $p-d,p,p+d$ are all primes. For examples, $(3,5,7)$ for $p=5$ and $(463,541,619)$ for the 100th prime ...
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292 views

The number of twin prime averages in the interval $[a,b]$ has a nice formula. What are some methods for analyzing such a formula?

Let $a = p_{n}+1, b = p_{n+1}^2 -1$ so that $[a,b]$ is a growing, moving interval. The number of twin prime averages in $[a,b]$ is given by: $f(a,b) = b - a + \sum_{1 \ \neq \ d \ \mid \ p_n\#} (-1)^{...
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121 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}^...
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193 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 ...
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137 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&...
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1answer
93 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 ...
7
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1answer
335 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, ...
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209 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.
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1answer
177 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 ...
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71 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 ...
18
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1answer
748 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 ...
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168 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。 ...
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48 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
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1answer
210 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 ...
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1answer
131 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 $...
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1answer
123 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 ...
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1answer
117 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....
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93 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 $\,...
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138 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 $[2n+1]^2$ ...
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1answer
212 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 ...
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103 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 ...
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2answers
401 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, ...
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327 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 ...
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1answer
198 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 ...
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1answer
669 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 ...
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1answer
903 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 $\,m(k)\,$ ...
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1answer
146 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 ...
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2answers
120 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\...
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1answer
224 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 ...
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98 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 ...
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3answers
582 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 ...
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1answer
255 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 ...
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1answer
446 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 ...
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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 ...
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1answer
195 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 ...
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1answer
171 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{...
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1answer
186 views

A Bonse's inequality for semiprimes, with a good mathematical content

A semiprime $s$ is a positive integer that is the product of two prime numbers, see Semiprine the encyclopedia Wikipedia. A well-known inequality, with applications, that involves prime numbers is the ...
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1answer
677 views

Can Scholze's perfectoid spaces bridge the gap for twin prime conjecture? [closed]

It seems than an analogue of the twin prime conjecture for polynomials in finite fields has been solved: see https://www.quantamagazine.org/big-question-about-primes-proved-in-small-number-systems-...
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275 views

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 : $$\#\{(...
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259 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}\...
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3answers
755 views

Positive proportion of logarithmic gaps between consecutive primes

For $x, \lambda > 0$, define $$S_\lambda(x) := \#\{p_{n+1} \leq x : p_{n+1} - p_n \geq \lambda \log x\} ,$$ where $p_n$ is the $n$th prime number. It is known [1] that an uniform version of the ...
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119 views

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 ...
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631 views

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 \...
5
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1answer
361 views

consecutive prime gaps and explicit bound

I am aware of the theorem that $p_{n+1}- p_n \leq n^{0.525}$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit "for all sufficiently large numbers" ...
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2answers
980 views

$\pi((n+1)^2)-\pi(n^2) \le \pi(n)$ for all $n \ge 370$?

There are some conjectures of the form: There always exist at least $X$ prime numbers between $A$ and $B$. Examples: Bertrand's postulate: for every $n>1$ there is always at least one prime $p$ ...
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123 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 ...
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1answer
432 views

Moments of merit

The merit of a prime gap equals $(p_{n+1}-p_n)/\ln p_n$. One can interrogate the statistics of merit by first restricting $n<M$ for some $M$, and then letting $M$ approach $\infty$. The very ...