All Questions
130 questions
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 ...
- 6,969
-2
votes
1
answer
396
views
Published articles in journals about the Firoozbakht's conjecture, whose main goal or focus is the study of this conjecture
I would like to know what articles are in the literature about the known as Firoozbakht's conjecture, see the Wikipedia Firoozbakht's conjecture.
Question. What articles have been published in ...
- 188k
0
votes
0
answers
83
views
Is it possible to get a conjecture similar to Mandl's conjecture for a different arithmetic function of number theory, mainly related to primes?
I'm curious to know if are in the literature analogous conjectures to the conjecture due to Mandl, I ask about these analogous conjectures for different sequences playing an important role in number ...
- 39.9k
4
votes
0
answers
273
views
Kaczorowski's Paper on Distribution of Primes
I am looking for a digital copy of the following paper by Jerzy Kaczorowski: ON THE DISTRIBUTION OF PRIMES (mod4)
https://www.degruyter.com/view/j/anly.1995.15.issue-2/anly.1995.15.2.159/anly.1995.15....
- 4,713
5
votes
0
answers
194
views
Asymptotic expansion for the average of $\omega(n)^2$
Let $\omega(n)$ be the prime factors counting function. I computed that for any $k\geq 0$, there exist certain constants $c_{-1},c_0,c_1,c_2,...c_k$ such that
$$\sum_{n\leq x}\omega(n)^2=x(\log\log x)...
5
votes
3
answers
809
views
Positive proportion of logarithmic gaps between consecutive primes
For $x, \lambda > 0$, define
$$S_\lambda(x) := \#\{p_{n+1} \leq x : p_{n+1} - p_n \geq \lambda \log x\} ,$$
where $p_n$ is the $n$th prime number. It is known [1] that an uniform version of the ...
- 114k
0
votes
1
answer
556
views
Order of magnitude of $\sum \frac{1}{\log^2{p}}$, or $\sum \frac{1}{\log^a{p}}$ for arbitrary power $a$ [closed]
In this MO question, it says that we have
$$ \sum_{p<n} \frac{1}{\log{p}} =\frac{n}{\log^2 n}+O\left(\frac{n\log\log n}{\log^3 n}\right).$$
where the sum is on all primes $p$, up to some max ...
- 1,322
1
vote
0
answers
274
views
On Primes in Arithmetic Progressions
I was wondering if the following approach is being attempted to prove the twin-prime conjecture.
Tao and Green proved in their paper (2006), that there are arbitrarily long arithmetic progressions ...
- 1
2
votes
0
answers
160
views
Where can I find a copy of this paper of Chowla and Vijayaraghavan?
Does anyone know where I can find a copy of Chowla and Vijayaraghavan's paper, ''On the largest prime divisors of numbers''?
The relevant literature say it was published in the Journal of the Indian ...
- 4,713
2
votes
0
answers
76
views
Is there an estimate available for a sum of the form $\sum_{\mathbf{x} \equiv \mathbf{a} (H) } \mu^2(x_1 x_2)$
I am interested in a sum of the shape
$$
\sum_{ \substack{ 1 \leq x_1, x_2 \leq B\\
\mathbf{x} \equiv \mathbf{a} (H) } } \mu^2(x_1 x_2).
$$
I figured it must have been considered before, but I have ...
- 3,625
26
votes
1
answer
1k
views
What is the status on this conjecture on arithmetic progressions of primes?
The Green-Tao theorem states that for every $n$, there is an arithmetic sequence of length $n$ consisting of primes.
For primes, $p$, let $P(p)$ be the maximum length of an arithmetic progression of ...
- 4,713
0
votes
0
answers
759
views
On sets of coprime integers in intervals
Briefly,
Question: Is it "good enough" to use least prime factor in choosing a maximal set of coprime integers in an interval?
The post title comes from a 1993 paper of Erdos and Sarkozy. They ...
- 13k
2
votes
1
answer
515
views
On comparing two almost injective divisor maps
Edit 2018.08.08 This answer https://mathoverflow.net/a/307881 will be updated to give recent information about S, especially a forthcoming preprint. End Edit 2018.08.08
In an introductory post on ...
CommunityBot
- 1
0
votes
0
answers
116
views
Reference request for bounds of $n$-th composite
Motivation
I will briefly elaborate here my motivations for asking the question. If you are not interested in it then please go to the questions.
Recently during trying to understand and prove the ...
CommunityBot
- 1
5
votes
1
answer
472
views
Is the following weak version of second Hardy-Littlewood conjecture already known?
Very recently I was going through my previous MSE posts and I stumbled upon some of them regarding the Second Hardy-Littlewood Conjecture which states that,
For all $x,y\ge 2$ we have, $$\pi(x)+\...
- 105k
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" ...
- 105k
3
votes
1
answer
276
views
Almost-Primes in Short Intervals
Let $S$ be the set of integers which are a product of $k$ distinct primes, $k$ a fixed positive integer (the condition that the primes are distinct is not crucial). Landau used the Prime Number ...
- 4,713
0
votes
0
answers
115
views
What is known about the lower bound for the integers $n$ for which $n$ minus the first $k$ odd primes are $k$ composite numbers?
Question edited in view of the comments below
By Yamada's paper we can conclude that if $n>e^{e^{36}}$ be an even number then it can always be written as the sum of a prime and a semi-prime.
My ...
CommunityBot
- 1
4
votes
1
answer
530
views
Reference for inequality for $\sum\limits_{d \mid n}\frac{\log d}{d}.$
Let $f(n)=\sum\limits_{d \mid n}\frac{\log d}{d}.$
It is not hard to see that $f(n)\ll(\log\log n)^2$. Is there any reference for this inequality?
EDT 1: A possible answer is Analysis of the ...
- 12.3k
17
votes
2
answers
2k
views
Is every odd positive integer of the form $P_{n+m}-P_n-P_m$?
I am looking for a comment, reference, remark, or proof of three conjectures as follows:
Conjecture 1: Let $x$ be an odd positive integer. Then there exist two integers $n, m \ge 2$ so that $$x=P_{n+...
CommunityBot
- 1
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$ ...
- 43.7k
4
votes
2
answers
1k
views
Primes $p$ for which $2p-1$ is prime
It's a well-known open problem (Sophie-Germain primes) whether there are infinitely many primes $p$, $2p+1$. What about $p$, $2p-1$?
Seemingly it's also an open problem (see here and the linked ...
- 105k
0
votes
1
answer
321
views
On the largest prime factor of $1+n^k$
For every positive integer $n>1$ , let $f(n)$ denote the largest prime factor of $n$. How fast does $f(1+n^k)$ grow with respect to $k$ ? Is it true that $f(1+n^k) > 2k, \forall n >2, \forall ...
- 44.4k
0
votes
0
answers
142
views
Mobius function on values of an irreducible quadratic polynomial
Are there infinitely many integers $n$ for which $n^2 + 1$ is square-free, and has an even number of (necessarily distinct) prime factors ?
- 11.3k
8
votes
2
answers
354
views
Let $f \in \mathbb{Z}[x]$. Does $\bar{f}$ have as many roots in $\mathbb{F}_p$ as $f$ has in $\mathbb{C}$ for infinitely many primes $p$?
Let $f \in \mathbb{Z}[x]$ be a nonconstant polynomial. Consider $\bar{f} \in \mathbb{F}_p[x].$ Let $\rho_p$ be the number of distinct roots of $\bar{f}$ in $\mathbb{F}_p$, and let $\rho$ be the number ...
- 50.6k
3
votes
0
answers
154
views
Is there a name for sequences of integers reduced to their lowest prime divisors?
When trying to obtain the value of Jacobsthal's function for some $n$; to find the largest sequence of consecutive numbers that are all coprime to $n$, one approach (and the only direct approach that ...
- 39.9k
0
votes
1
answer
306
views
Have you seen this prime distribution before?
The basic question is : has this system been considered before, and how do I find it? References to the literature would be most welcome, but I am asking for reasonable search terms. I will try the ...
- 39.9k
6
votes
1
answer
1k
views
Reference request: Dickman, On the frequency of numbers containing prime factors
I've been trying without success to find the paper
Dickman, Karl, "On the frequency of numbers containing prime factors of a certain relative magnitude." Ark Mal., Astronomi och Physik, 22A (10), ...
- 2,992
8
votes
1
answer
811
views
Primes of the form $x^2 + y^2 + 1$
There are infinitely many primes of the form $x^2+y^2+1$, as proved by Bredihin. Motohashi improved the result by showing that there were $\gg x/\log^2 x$ such primes up to $x$. But we expect $\Theta(...
- 43.7k
19
votes
2
answers
2k
views
Who first proved the generalization of Bertrand's postulate to (2n,3n) and (3n,4n)?
In Wikipedia's page for Bertrand's postulate, it is said that its (2n,3n) version was proved by El Bachraoui in 2006. Seems likely that it was first proved way before than that! Can anyone point to ...
- 14.6k
2
votes
2
answers
393
views
Playing leapfrog with primes
In connection with how primes jump (How do these primes jump?),
I consider the following game.
Let $R$ be a finite set of positive integers. For this question, I content myself with $R$ being the $k$ ...
- 151k
6
votes
1
answer
360
views
Friable Numbers In Short Intervals: Density Estimates?
I am hoping for explicit numerical estimates like the following sample (with made up numbers, though it might be true): for every $n \gt 10^6$ and every $b$ with $b^2 \lt n \lt b^3$, the number of ...
CommunityBot
- 1
2
votes
2
answers
338
views
Weak form of Brocard's conjecture
I ask this out of curiosity, motivated by a question asked by one of my students.
The Brocard's conjecture claims that there exist at least four prime numbers between $p_{i}^2$ and $p_{i+1}^2$, where ...
- 6,984
3
votes
0
answers
408
views
The second conjecture about the degrees of special polynomials
Define the congruence "modulo m" on exponential Taylor series following the previous post (A conjecture about the degrees of special polynomials)
It has been conjectured, that if we define the ...
- 237
3
votes
1
answer
224
views
PNT analog for primes inside a structured set
Let $\Bbb T$ be the set of all square free integers with ordering derived from $\Bbb N$. Essentially $PNT$ says if you pick $\log N$ integers less than $N$ you can expect one of them to be prime.
...
- 13.9k
7
votes
1
answer
382
views
$\log \log p / \log \log n$, where $p|n$, gets equidistributed in [0,1] (for almost all $n$)
According to Hardy-Ramanujan/Erdős-Kac we know that usually there are $\sim\log\log n$ prime numbers in a factorization. But if you pick up a natural number at random, and you factor it, what is the ...
- 105k
17
votes
0
answers
891
views
An elementary proof that, for every fixed $n \in \mathbf N^+$, there are infinitely many primes $\equiv -1 \bmod n$
This morning, I made a comment to a comment to a question of Ayman Moussa, only to point out that, among many others, there is an elementary proof of Dirichlet's theorem on the existence of infinitely ...
- 10.5k
6
votes
2
answers
754
views
ASCII prime plots and prime-rich quadratic polynomials
This is a series of questions inspired by the MathOverflow question
Find the least prime so that p-1 has two factors greater than $m$ and $n$ posted by Aaron Sterling.
I suggested plotting primes by ...
CommunityBot
- 1
5
votes
1
answer
259
views
Central binomial coefficients deprived of $2$'s: not radicals?
In the paper, P Erdos, R Graham, I Ruzsa, E Straus, On the prime factors of $\binom{2n}n$, Math. Comp., 29:83–92, 1975, it was conjectured that the central binomials are never square-free for $n>4$....
- 43.7k
5
votes
1
answer
414
views
Primality test for $2p+1$
In 1750 Euler stated following theorem :
Let $p \equiv 3 \pmod 4$ be prime then $2p+1$ is prime iff $2p+1 \mid 2^p-1$ .
In 1775 Lagrange gave a proof of the theorem .
Recently I have formulated ...
1
vote
1
answer
203
views
Best bound on $p, p+2k$ with $k$ fixed
Given some integer $k>0$, there are $O(x/\log^2 x)$ primes $p \le x$ such that $p+2k$ is also prime. It has been conjectured at least since Hardy-Littlewood that
$$
\pi_{2k}(x) \sim c_{2k}\int_2^x\...
- 105k
2
votes
1
answer
450
views
Is there a "small $\omega$" number theorem?
In my studies of how primes jump (search this forum for a link), a question has been raised which may have been studied. Can anyone jump-start my literature search with references regarding the ...
4
votes
1
answer
647
views
Did Erdős prove there are two primes $4a+1, 4b+3$ between between $n$ and $2n$?
http://mathworld.wolfram.com/ChoquetTheory.html
Is the claim in the link true? Here's the reference given there:
https://www.renyi.hu/~p_erdos/1934-01.pdf
Erdős proved that there exist at least one ...
CommunityBot
- 1
4
votes
0
answers
176
views
Are there any results about this higher degree Titchmarsh divisor problem?
Does there exist an asymptotic formula for $\sum_{p\le x}\tau(p−1)^n$ ? Here $n$ is an arbitrary positive integer and $\tau$ is the divisor function. The case of $n=1$ was done by Linnik, but when $n$ ...
- 105k
1
vote
3
answers
286
views
Reference book for primality testing [closed]
im searching for good reference to understand the primality testing idea
especially the Elliptic curves and primality for stirling numbers first and second ones , so can any one suggest to me good ...
- 199
1
vote
1
answer
231
views
An estimation of $p_n$
There seems to exist an asymptotic line
$$\displaystyle a+bx\sim \frac{x e^x}{p_{n}-x e^x}\; ,\;n=\lfloor e^x\rfloor\tag1$$
Which suggests an estimation
$$\displaystyle g(n)=\Big(1+\frac{1}{a+b\ln ...
CommunityBot
- 1
1
vote
0
answers
165
views
Reference request: Bounding exponential sum $\sum_{x \in [0,X]} \Lambda(x) e(\beta_d x^d + \ldots + \beta_1 x )$
Let $1 \leq i \leq d$, $q \in \mathbb{N}$, and $0 \leq a_{i} < q$. Let
$$
\mathfrak{M}^{(i)}_{a_{ i}, q} (C) =\{ \beta_{i} \in [0,1) : | \beta_{i} - a_{i}/q | \leq (\log X)^{C} X^{-i} \} .
$$
We ...
- 3,625
2
votes
0
answers
149
views
$f(x)$-th largest number of prime factors
Given a sufficiently well-behaved function $1\le f(x)\le x$ and a multiset $S=\{\omega(n): 1\le n\le x\}$, what can be said about the asymptotics of the $f(x)$-th largest member of $S$? In other words,...
- 9,114
2
votes
0
answers
236
views
On the cardinality of the set of right-truncatable primes
We say that the (base ten) prime number $p=a_{n}a_{n-1}a_{n-2}\cdots a_{1}a_{0}$ is right-truncatable if all of the following numbers are prime:
\begin{eqnarray*}a_{n},\\a_{n}a_{n-1},\\ a_{n}a_{n-1}...
- 8,842
13
votes
1
answer
333
views
Elementary prime-generating sequences
A student of mine keeps coming again and again and telling "I've found a formula $n\mapsto f(n)$ giving all primes" or sometimes "infinitely many primes", where $f$ is a classical function (I mean ...
- 13k