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Primes generated by cyclotomic polynomials

Let $p$ be an odd prime, and let $f=\Phi_p$ be the $p$-th cyclotomic polynomial. Denote by $S_p$ the set of primes $q$ such that there exists a sequence of primes $p_1,\dots, p_g$ such that $p_1=f(1)=...
Maurizio Moreschi's user avatar
0 votes
0 answers
75 views

Existence of smooth integers in every residue class with large modulus

Let us say that a positive integer $x$ is $y$-power smooth, if the largest prime power divisor of $x$ is at most $y$. In what follows, let $C$ be any real number larger than $1$ and, for an integer $x$...
Woett's user avatar
  • 1,663
5 votes
1 answer
310 views

Is there a statement in Presburger arithmetic about primes this simple heuristic fails for?

I came up with the following conjecture while thinking about ways to formulate some heuristics about primes: Conjecture: Given a statement $s$ in Presburger arithmetic, using an additional unary ...
Daniel Weber's user avatar
  • 3,319
5 votes
0 answers
131 views

Taking integer values of a sequence of Beurling primes

Let $P=(p_j)_{j=1}^\infty$ be an increasing sequence of real numbers with $1<p_1$ and $\lim_{j\to\infty}p_j=\infty$. As mentioned in [1], Beurling proved that if the multiplicative group $N_P$ ...
Anon12345's user avatar
0 votes
1 answer
292 views

Factorization trees and (continued) fractions?

This question is inspired by trying to understand the lexicographic sorting of the natural numbers in the fractal at this question: Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , ...
mathoverflowUser's user avatar
5 votes
1 answer
737 views

Smallest prime factor of numbers

The literature refers to smooth integers as \begin{equation}\Psi(x,y):=\#\{n\le x:P_1(n)\le y\},\end{equation} where $P_1(n)$ is the largest prime factor of $n$. There are lots of results studying $\...
user avatar
1 vote
2 answers
390 views

Solving a recurrence relation for the prime counting function?

I have found some number sequence $c_n = 1+b_n$ for $n \ge 0$, where $b_n = $ A307977(n). I am trying to solve the following recurrence relation for the prime counting function: $$\forall n \ge 3: \pi(...
mathoverflowUser's user avatar
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 ...
Đào Thanh Oai's user avatar
2 votes
0 answers
103 views

On equidistribution of primes in positive characteristic

In S. Lang's book "Algebraic Number Theory" (1986), page 317, Theorem 6 states essentially that given $P$ a set of primes, let $\tau:P\longrightarrow J$ be the typical idèle map taking ...
Hair80's user avatar
  • 675
1 vote
0 answers
132 views

Are the binary digits of the sequence of the prime numbers correlated?

Let $p_n\geq 3$ be the $n$th prime number with the binary expansion $p_n = \sum_{k=0}^{\infty} b_{nk}2^k$ ($b_{nk}\in\{0,1\}$). Let's write $q_{nk} = 1-2b_{nk}$. Question: Is it true that for $k,l\...
Onur Oktay's user avatar
  • 2,605
5 votes
1 answer
340 views

About an asymptotic behavior in number theory

Where can I read about the asymptotic behavior (with $N$ tending to infinity) of the sum of the fractional parts obtained from dividing $N$ by all prime numbers up to $N$ divided by the number of ...
26 votes
0 answers
567 views

Elliptic analogue of primes of the form $x^2 + 1$

I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but ...
Marty's user avatar
  • 13.3k
4 votes
1 answer
601 views

Reference for a proof of Euclid's Theorem for the infinitude of primes

I would be curious to have a reference for the following proof of Euclid's Theorem on the infinitude of primes: Using Legendre's formula (also called de Polignac's formula) for $p$-adic valuations of ...
Roland Bacher's user avatar
1 vote
0 answers
98 views

Reference request for a result in additive combinatorics

Let $p$ be a prime number and $[p-1]=\{1, 2, \ldots, p-1\}$. The following proposition is proved: (but I cannot find out where) Proposition: The non-empty subset sums of $[p-1]$ are equally ...
Konstantinos Gaitanas's user avatar
2 votes
1 answer
305 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 ...
soliton's user avatar
  • 149
3 votes
1 answer
228 views

What fraction of the values of a quadratic polynomial can be prime?

I have an explicit, monic quadratic polynomial $P(x)$ and an integer $m$. Can I bound the number of prime values in $P(0), P(1), \ldots, P(m)$? A reference would be appreciated, if available. An ...
Charles's user avatar
  • 9,114
1 vote
1 answer
153 views

Number of distinct near-squares primes dividing an odd perfect number

I'm curious about if the following question is in the literature or what work can be done about it. Denote the number of distinct primes dividing an odd perfect number $N$ with the arithmetic function ...
user142929's user avatar
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 \...
Eric Naslund's user avatar
  • 11.4k
3 votes
0 answers
158 views

What can be said about the primality of Zsigmondy numbers?

I am cross-posting this from math.stackexchange, as it has received upvotes but no comments/answers after a couple months. Let $\mathcal{Z}(n,a,b)=\frac{\Phi_n(a,b)}{\gcd (\Phi_n(a,b),n)}$ be the $n$-...
Tejas Rao's user avatar
  • 101
3 votes
1 answer
418 views

Counting cubic residues mod p

Given a prime $p=3m+1$, $(p-1)/3$ of the residues mod $p$ are cubic residues. So heuristically, for any given integer $k>1$ not a perfect cube, we would expect that about 1/3 of the primes $\equiv1\...
Charles's user avatar
  • 9,114
14 votes
1 answer
424 views

Unpublished result of Rosser in Sieve Methods book

Erdős and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert. ...
Bjørn Kjos-Hanssen's user avatar
9 votes
2 answers
547 views

Primes between $x$ and $x+x^\theta$

Iwaniec [1] proved that $$ \pi(x+x^\theta)-\pi(x) < \frac{(2+\varepsilon)x^\theta}{\eta(\theta)\log x},\ x>x_0(\varepsilon,\theta). $$ with $$ \eta(\theta)=\frac{15\theta-2}{9}. $$ (Actually, he ...
Charles's user avatar
  • 9,114
25 votes
1 answer
911 views

Reference request for a proof of the two-square Theorem

One can show (see below for a sketch of a proof) that every odd prime number $p$ can be written in exactly $(p+1)/2$ different ways as $$p=a\cdot b+c\cdot d$$ with $a,b,c,d\in\mathbb N$ satisfying $\...
Roland Bacher's user avatar
1 vote
0 answers
139 views

Alternative Mersenne numbers

Let $\ b\in\mathbb Z,\ $ and $\ |b|>1.\ $ Call $$ M_b(n)\ :=\ \frac{b^n-1}{b-1} $$ to be $n$-th Mersenne number mod $b$. The necessary condition for $\ M_b(n)\ $ to be a prime is that $\ n\ $ is a ...
Wlod AA's user avatar
  • 4,786
3 votes
0 answers
154 views

Reference request for the following results

I am looking for references on the following results. In what follows $\pi(x)$ denotes the prime counting function. Result 1. For all real $k>1$ there exists $x^k_0 \in \mathbb{R}$ such that for ...
S. Das's user avatar
  • 31
6 votes
0 answers
149 views

Dickson's conjecture for Beatty sequences

A particular case of Dickson's Conjecture states that for $a_1,q_1,a_2,q_2$ with $(a_1,q_1)=(a_2,q_2)=1$, there are infinitely many $n$ for which $q_1 n + a_1$ and $q_2 n+a_2$ are both prime, provided ...
Joshua Stucky's user avatar
3 votes
1 answer
329 views

Fully explicit Linnik's Theorem

Linnik's Theorem states that there exist absolute constants $c$ and $L$ such that for every $m \in \mathbb{N}$ and every $a$ coprime to $m$, there is a prime $p$ with $p \equiv a \pmod{m}$ and $p < ...
Woett's user avatar
  • 1,663
8 votes
2 answers
1k views

Question about functions $f: \mathbb{Z}^+ \to \mathbb{Z}^+$ such that $x$ is prime whenever $f(x)$ is prime

Let $f: \mathbb{\mathbb{Z}^+} \to \mathbb{Z^+}$ be a function and suppose $(\star)$ For all integers $x \geq 3$, if $f(x)$ is prime, then $x$ is prime. A trivial example of such a function is the ...
matt stokes's user avatar
6 votes
2 answers
804 views

Must Mersenne numbers be divisible by arbitrary large primes with exponent one?

Let $M_n$ denote the Mersenne numbers $M_n=2^n-1$. As $n$ varies, must $M_n$ be divisible by arbitrary large prime $p$ with exponent one, i.e. $p \mid M_n, p^2 \nmid M_n$? In other words, must the ...
joro's user avatar
  • 25.4k
3 votes
1 answer
247 views

Explicit bounds on number of squarefree numbers coprime to a certain number

We know that the number of squarefree integers $\le x$ that are coprime to $A$ is $$ Q_A(x) = x \prod_{p|A} \left(1-\frac{1}{p}\right) \prod_{p \nmid A} \left(1-\frac{1}{p^2}\right) + O(\sqrt{x}). $$ ...
Iguana's user avatar
  • 301
20 votes
2 answers
4k views

information-theoretic derivation of the prime number theorem

Motivation: While going through a couple interesting papers on the Physics of the Riemann Hypothesis [1] and the Minimum Description Length Principle [2], a derivation(not a proof) of the Prime Number ...
Aidan Rocke's user avatar
  • 3,871
9 votes
1 answer
388 views

$π(x+y) - π(x) ≤ c·y/\ln(y)$ for some constant $c$?

(I posted this question on Math SE but it has had no answer for a year now so I would like to ask if anyone here can provide one.) Thinking about the prime number theorem, I wondered whether it is ...
user21820's user avatar
  • 2,912
12 votes
1 answer
526 views

Equidistribution of $\{\alpha p\}$ for $p$ in an arithmetic progression

Let $\alpha$ be irrational. A famous theorem of Vinogradov says that $\{ \alpha p\}$ is equidistributed in $[0,1]$ as $p$ runs over all primes. Let $a,q$ be natural numbers with $\gcd(a,q) = 1$. Then ...
Daniel Loughran's user avatar
2 votes
0 answers
313 views

On the Chowla and twin prime conjectures

I'm reading https://terrytao.wordpress.com/tag/chowlas-conjecture/ and at some point it is mentioned that, the twin prime conjecture is a variant of Chowla's conjecture that $\sum_{n\leq x} \lambda(n)\...
Q_p's user avatar
  • 1,019
0 votes
1 answer
204 views

On $(\prod_{\substack{1\leq s\leq X\\s\text{ semiprime}}}s)(\sum_{\substack{1\leq s\leq X\\s\text{ semiprime}}}\frac{1}{s})$ as $X\to\infty$

Few weeks ago an user from Mathematics Stack Exchange answered my question On an inequality that involves products and sums related to the sequence of semiprimes (asked May 26). It seems that for ...
user142929's user avatar
1 vote
1 answer
177 views

Arithmetic progressions, given a prime

I have recently become interested in reading a little more on certain directions regarding primes in arithmetic progressions (AP). I would appreciate specific paper references (with the journal and ...
user avatar
5 votes
4 answers
819 views

Can one show combinatorially how $\operatorname{lcm}(1, \dotsc, n)$ grows?

Let us write $M(n)$ for $\operatorname{lcm}(1,\dotsc,n)$ for $n$ a positive integer. Asymptotically $M(n)$ tends toward $e^n$. This result uses analytic number theory. (Lcm is least common multiple, ...
Gerhard Paseman's user avatar
5 votes
2 answers
1k views

Error term in Mertens' third theorem

Mertens' third theorem states that: $$\prod_{\substack{ p \leq x \\ \text{p prime} }} \left( 1 - \dfrac{1}{p} \right) \sim \dfrac{e^{-\gamma}}{\log(x)}$$ Question: what is the best functions (...
Lagrida Yassine's user avatar
4 votes
1 answer
291 views

Reference / Survey for finite field analog number theory

It is folklore that many number theoretic results on prime numbers have a simpler-to-prove finite field analog. For example, on the one hand, the proof of the Prime Number Theorem $$\#\{\text{prime ...
peter's user avatar
  • 43
4 votes
1 answer
332 views

Estimating certain short Kloosterman sums

Recall that for the classical Kloosterman sum $$ K(a,b,p^t):= \sum_{x \in (\mathbb{Z}/ p^t \mathbb{Z})^* } \psi \left(\frac{ax+bx^{-1}}{p^t} \right),$$ where $\psi(x)=e^{2\pi ix}$, $a,b,t$ are natural ...
JACK's user avatar
  • 421
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 ...
Safwane's user avatar
  • 1,197
2 votes
1 answer
1k views

Sum of the digits in base $p+1$

Definition Let $W$ be the function , defined as $W(a,b)=r$ given $a,b\in \mathbb{Z_+}$ and $a>1$ Take $m$ to be the integer s.t. $a^{m+1} \ge b > a^{m}$, i.e. $m = \...
Pruthviraj's user avatar
-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 ...
user142929's user avatar
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 ...
user142929's user avatar
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....
primefinder's user avatar
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)...
The Number Theorist's user avatar
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 ...
Kello's user avatar
  • 113
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 ...
Mike Battaglia's user avatar
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 ...
user136205's user avatar
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 ...
Q_p's user avatar
  • 1,019