All Questions
36 questions
20
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 ...
- 35.5k
20
votes
2
answers
2k
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 ...
- 5,470
12
votes
2
answers
1k
views
Prime differences and zero multiplicity
Concerning gaps between consecutive primes, Paul Erdős conjectured that:
$$\sum_{p_n < x} (p_n -p_{n-1})^2 = O(x \log x)$$
Let's call this hypothesis EH. Assuming the Riemann hypothesis (RH), ...
- 232
10
votes
0
answers
350
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 ...
- 533
9
votes
1
answer
400
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
$$\pi(x+y)-\pi(x)\gg \...
- 11.4k
8
votes
1
answer
471
views
Conjecture about the density of primes
Conjecture
For any sufficiently large integer $kn$ , the sequence representing
the number of primes in each block obtained by splitting $kn$ into $k$
equal blocks, is a strictly decreasing sequence, ...
- 197
7
votes
1
answer
1k
views
Some interesting experimental results about the distribution of primes
Let's consider the following metric of the gap between consecutive primes
$$m(k)=\frac {p_k^2-p_{k-1}^2} {24}\;\;\;\;\;(k\ge4)$$
Now, let's define the function
$\delta(k)=m(k)\;\;\;\;$ if $\,m(k)\,$ ...
- 1,008
7
votes
1
answer
481
views
Some conjectures about prime gaps
I checked some relations between primes, here $1<n<10^5$ and $p_n$ is the $n$th prime.
$a) p_n^{1/3} - p_{n-1}^{1/3}<1/2$
$b) p_n^{1/n} - p_{n-1}^{1/n}<1/n $
$c) (\log p_n)^{1/2} - (\...
- 521
7
votes
1
answer
596
views
What consequences would follow from the density hypothesis?
Let $N(\sigma,T)$ denote the number of zeros $\rho=\beta+\gamma i$ of the Riemann zeta function satisfying $\beta\ge \sigma$ and $0<\gamma\le T$, counted with multiplicity. Then the "Density ...
- 435
6
votes
2
answers
1k
views
$\pi((n+1)^2)-\pi(n^2) \le \pi(n)$ for all $n \ge 370$?
There are some conjectures of the form: There always exist at least $X$ prime numbers between $A$ and $B$. Examples:
Bertrand's postulate: for every $n>1$ there is always at least one prime $p$ ...
- 1,776
6
votes
1
answer
903
views
How to explain this prime gap bias around last digits?
My question is related to this article by Oliver and Soundararajan (article about a bias in the distribution of the last digits of consecutive prime numbers).
After trying some python experimental ...
6
votes
2
answers
837
views
A generalization of strong primes
In this post we denote the sequence of prime numbers as $p_k$ for integers $k\geq 1$. I don't know if the following definition is in the literature.
Definition. We define the $\theta$-strong primes, ...
6
votes
0
answers
465
views
On improvements of the GPY sieve
When $\chi_\mathbb P(n)$ denotes the characteristic function of primes and $\mathcal H=\{h_1,h_2,\dots,h_k\}$ is some admissible $k$-tuple, the GPY sieve can be formulated as follows:
$$
S(x)=\sum_{x&...
- 1,315
5
votes
2
answers
1k
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 $C$ ...
- 1,018
5
votes
3
answers
809
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 ...
- 113
5
votes
1
answer
458
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 ...
- 17.6k
5
votes
1
answer
434
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" ...
user95470
5
votes
0
answers
326
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 ...
- 3,098
5
votes
0
answers
340
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 ...
4
votes
1
answer
507
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 ...
- 1,197
4
votes
0
answers
335
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 $g_n$ or $g(p_n)$ is the difference between the $(n+1)$-st and the $n$-th prime numbers. Using my ...
- 1,776
3
votes
0
answers
292
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 ...
- 791
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 ...
- 71
2
votes
1
answer
545
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 ...
- 13.9k
2
votes
1
answer
170
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 ...
user145059
2
votes
0
answers
114
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 $\,...
- 1,008
1
vote
0
answers
240
views
Liu's new sieve weight
Does Liu's sieve weight (in his arXiv paper "On the gap between primes")
$$sieve(n)=(\sum_{\substack{d_i\mid (n-h_i),i=1,\cdots,k\\ (d_1,\cdots,d_k)\in\mathcal{D}}}\lambda_{d_1,\cdots,d_k} ...
- 11
1
vote
0
answers
277
views
Prime numbers in this region
Let $q \geq 5$ be a prime number, and consider : $N_q = \displaystyle{\small \prod_{\substack{p \leq q \\ \text{p prime}}} {\normalsize p}}$
Using Chinese remainder theorem we can show that :
$$\#\{(...
- 521
0
votes
2
answers
153
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\...
- 6,974
0
votes
1
answer
137
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....
- 1,008
0
votes
0
answers
50
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 ...
- 29
0
votes
0
answers
106
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 ...
-1
votes
1
answer
246
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 ...
-3
votes
1
answer
237
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 ...
- 233
-4
votes
3
answers
670
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 ...
-4
votes
1
answer
229
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 ...
- 1,776