Questions tagged [prime-numbers]

Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.

Filter by
Sorted by
Tagged with
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
3 votes
0 answers
79 views

Are numbers which are the product of n primes more common than numbers which are the product of n-1 primes? [duplicate]

In a recent video (https://www.facebook.com/188916357807416/videos/519169035700435/) Stephen Wolfram wonders whether, for every integer n>2, eventually the number of integers which are precisely the ...
Bernardo Recamán Santos's user avatar
4 votes
1 answer
269 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
6 votes
1 answer
855 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 ...
Thierry Boulord's user avatar
1 vote
1 answer
131 views

Another kind of primality related to tessellations by polygons

You can define a number $p$ to be prime by "no tessellation of $p$ identical squares forms a convex figure". This suggests what I'll call a t-prime $p$, defined by "no tessellation of $p$ identical ...
Eric Braude's user avatar
0 votes
0 answers
90 views

Set of primes $p_{1}\equiv 3 \bmod p_{2}$ such that $\phi(2^{\frac{{p_1}-3}{p_2}}-1)\equiv 0 \bmod p_1$ with $p_1,p_2\equiv 3\bmod 4$?

let $p_1$ and $p_2$ be positive primes such that $p_1,p_2 \equiv 3\bmod 4$ and $\phi$ is the Euler totiont function , I want to find the Set of primes $p_{1}\equiv 3 \bmod p_{2}$ such that $\phi(2^...
zeraoulia rafik's user avatar
2 votes
1 answer
225 views

How does one prove that the density of unusual numbers is $\ln 2$?

The Wikipedia page for unusual number states that the density of that set is $\ln 2$, and that this was proven by Schroeppel in 1972. The only source that I found for that is the HAKMEM document, and ...
Andrei Sipoș's user avatar
4 votes
1 answer
211 views

Primes in arithmetic progressions above a given threshold

Given co-prime $a,b$, Dirichlet's theorem states that there are infinitely many primes in the arithmetic progression $M = \{ a + bn : n \in \mathbb N\}$. Linnik's theorem asserts that the first such ...
Christoph Haase's user avatar
1 vote
1 answer
352 views

Equation of the Chebyshev $\psi$ function

Consider $\Psi(x)$ to be the Chebyshev function given by $$\Psi(x)=\sum_{n\leq x} \Lambda(n)$$ where $\Lambda(n)$ is the Mangoldt function which is equal 0 unless $n $ is prime power, and let $(E)$ ...
Aster Phoenix's user avatar
1 vote
0 answers
102 views

Write $p^2$ as $x^2+2y^2+3\times 2^z$ with $x,y,z$ nonnegative integers

In April 2018, I noted that the first integer $n>1$ with $n^2\not\in\{x^2+2y^2+3\times 2^z:\ x,y,z=0,1,2,\ldots\}$ is $$5884015571=7\times17\times49445509.$$ Question. Is it true that for each ...
Zhi-Wei Sun's user avatar
  • 14.4k
-1 votes
1 answer
137 views

Sufficient conditions on $ a_i,b_i$ for $a_1\phi(n)+b_1, \cdots, a_k\phi(n)+b_k$ to be simultaneously prime infinitely often?

I am really interested in sufficient conditions on $a_i, b_i$ guaranteeing that the linear forms $a_1\phi(n)+b_1,\dots, a_k\phi(n)+b_k$ become simultaneously prime for infinitely many positive ...
zeraoulia rafik's user avatar
2 votes
0 answers
81 views

quadratic residues and cubic polynomials [closed]

I'm really not sure about this, but I've heard somewhere that for any prime $p$, $|\sum_{x=0}^{p-1} (\frac{ax^3 +bx^2 +cx +d}{p} ) |\le \sqrt{2p}$ holds. Does anyone know a proof for this inequality ...
Junsukim's user avatar
  • 141
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 $\,m(k)\,$ ...
Augusto Santi's user avatar
2 votes
1 answer
152 views

Sequence of least prime-multiples with smallest Hamming weight

It is known that Almost all primes have a multiple of small Hamming weight, which makes me wonder what is known about the least multiples of primes that have least Hamming weight. Questions: what ...
Manfred Weis's user avatar
  • 12.6k
0 votes
0 answers
128 views

A quadratic trinomial that generates only prime numbers of the form $4m+1$

It is known that Euler's polynomials $\,n^2+n+p\,$ ($p\,$ prime) represent a prime for $\,n=0,\,...,\,p-2\,$ if and only if the field $\,Q (\sqrt{1-4p})\,$ has class number $\,h=1$. The best ...
Augusto Santi's user avatar
6 votes
1 answer
227 views

Is there a connection between the average 'compositeness' of a rational number and $\phi$ (golden ratio)?

Let $n\in N$, where $n = p_{1}^{k_{1}}p_{2}^{k_{2}}...p_{m}^{k_{m}}$ for $p_{i}$ prime. Define the 'density' of $n$ as: $d(n) = \frac{(p_{1}+1)^{k_{1}}(p_{2}+1)^{k_{2}}...(p_{m}+1)^{k_{m}}}{n}$ ...
swami's user avatar
  • 369
0 votes
1 answer
159 views

does the ratio of the count of rational numbers on an $n\times n$ grid to $n^2$, converge as $n$ tends to infinity [closed]

Suppose we order the rational numbers using the diagonal method (used to prove they are countable) using an $n\times n$ grid. Now suppose we count the distinct rational numbers (those points on the ...
swami's user avatar
  • 369
0 votes
1 answer
237 views

What can this $\int_{0}^{t} (\pi(x)-Li(x)) dx$ tell us about primes distribution?

Many papers I have read which are related to primes distribution only it discussed sign and refinement Bounds of $\pi(x)-Li (x)$ with $\pi(x)$ is a prime counting function and $Li (x)$ is the ...
zeraoulia rafik's user avatar
1 vote
1 answer
188 views

Resolution of an inequality on integers

I’m trying to resolve respect to $k$ the following inequality, $$ k\left(\log k +\log \log k-\alpha+O\left(\frac{\log \log k}{\log k}\right)\right)\geq x, $$ in order to obtain, under the condition $...
syazo's user avatar
  • 13
1 vote
0 answers
142 views

About the distribution of Fibonacci numbers that are primes

Let's consider the Fibonacci sequence, that is the sequence of naturals defined by: $F_1=F_2=1$ $F_{n+1}=F_{n}+F_{n-1}$ It is an open problem whether the sequence contains an infinite number of ...
Augusto Santi's user avatar
4 votes
0 answers
143 views

Moments of the prime counting function given the moments of the second Chebyshev function

I have read this article (Montgomery and Soundararajan: Primes in short intervals. http://arxiv.org/abs/math/0409258 ). In the second page of the article, it is stated that the mean and variance of $\...
carlorop's user avatar
  • 141
3 votes
1 answer
176 views

About the sum $S(p_n)=\sum_{1\le k\lt n}\,p_n\mod\;p_k$

For $\,p_n\gt2\,$ let's define the sum $\,S(p_n)=\sum_{1\le k\lt n}\,p_n\;mod\;p_k$, where $\,p_k\,$ represents the $\,k$-th prime. The first terms of the sequence $\,S(p_n)\,$ (OEIS A033955 - sum of ...
Augusto Santi's user avatar
1 vote
0 answers
175 views

Given a prime $\,p\ne3$, is it always possible to find another prime q such that $\,\phi(q)=\phi(p\cdot2^{2n+1})\,$ for some $n\in\mathbb{N}$?

Given a prime $\,p\ne3$, is it always possible to find another prime q such that $\,\phi(q)=\phi(p\cdot2^{2n+1})\,$ for some $n\in\mathbb{N}$ ($\,\phi\,$ is the Euler's totient function)? Some ...
Augusto Santi's user avatar
3 votes
1 answer
310 views

Why is this sequence a good prime-generator?

For $n \in \mathbb N$ we can observe the $n$ remainders $b_1,...,b_n$ by writing $n$ as $n=a_k \cdot k+b_k$ for $1 \leq k \leq n$. Because of the familiar division-with-remainder theorem we have $0 \...
user avatar
7 votes
1 answer
493 views

About semiprimal representations of $1$

Conjecture $A_1$ : For every $m \in \mathbb N \setminus \{1\}$ there exist mutually different primes $p_{r_1},...,p_{r_m}$ and mutually different primes $b_{w_1},...,b_{w_m}$ and numbers $i_1,...,i_m \...
user avatar
-1 votes
1 answer
281 views

Is it possible to determine whether the sequence $\,a_0=p,\;a_{n+1}=(a_n-2)\cdot a_n+2\,$ will reach another prime number?

Given a prime $\,p\,$ let's consider the following sequence: $a_0=p$ $a_{n+1}=(a_n-2)\cdot a_n+2$ Is it possible to determine whether the sequence $\,a_n\,$ will reach, sooner or later, another ...
Augusto Santi's user avatar
11 votes
1 answer
421 views

How many numbers $\le x$ can be factorized into three numbers which form the sides of a triangle?

Note: Posting in MO since it was unanswered in MSE Definition: We say that a natural number $n$ has triangular divisors if it has at least one triplet of divisors $n = d_1d_2d_3, 1 \le d_1 \le d_2 \...
Nilotpal Kanti Sinha's user avatar
0 votes
0 answers
141 views

Given $\,m=\prod_k {p_k}^{\alpha_k}\,$ and the function $\,g(m)=\sum_k \alpha_k(p_k-1)^2$, find all solutions of the equation $\,g(2n)=n$

Let's consider the unique decomposition of a natural number $\,m\,$ into its prime factors: $$\prod_k {p_k}^{\alpha_k}$$ Then, let's define the following arithmetic function (completely additive) $\,g:...
Augusto Santi's user avatar
13 votes
1 answer
1k views

About the number of primes which are the sum of 3 consecutive primes (OEIS A034962)

I made some numerical simulations about the number of primes which are the sum of 3 consecutive primes (OEIS A034962), that is for instance: $$5+7+11=23$$ $$7+11+13=31$$ $$11+13+17=41$$ $$17+19+23=59$$...
Augusto Santi's user avatar
1 vote
1 answer
116 views

Sequences of positive integers $(a_{k})_{k \in \omega}$ that only give finitely many zeros modulo $p_{k}$ in total for all polynomials

Let $(a_{k})_{k \in \omega}$ be a sequence of positive integers such that $a_{k} < p_{k}$, $a_{k} \leq a_{k+1}$ and $\lim_{k \rightarrow \infty} a_{k}=\infty$ where $p_{k}$ is the k-th prime ...
Florian Felix's user avatar
2 votes
0 answers
170 views

Funny questions about Moebius Function

I need to firstly claim that my research is not about number theory, however, I am pretty interested in it, especially funny questions in number theory, e.g. Kollatz Conjecture. Three years ago, I ...
cheng's user avatar
  • 41
0 votes
0 answers
77 views

Construction of (general class of) function(s), which sieves out primes, w.r.t. given conditions:

Consider the function $F(x)$ defined in following manner: $F(n)$ is finite (likely $F(x)\in[0,1]$) if $n$ is prime and zero otherwise: It has to satisfy following conditions: (1) $F(x)$ is ...
bambi's user avatar
  • 375
3 votes
1 answer
266 views

Solutions in primes of the equation $\,3p^2+q^2=r^2+3$

Let's consider the Diophantine equation $\,3p^2+q^2=r^2+3$. Actually, I am interested only in the solutions represented by sets $\,(p,q,r)\,$ of prime numbers. It's easy to prove that if $\,(p,q)\,$ ...
Augusto Santi's user avatar
4 votes
1 answer
313 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
2 votes
1 answer
161 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 ...
user avatar
7 votes
2 answers
905 views

Regularizing the sum of all primes

In the spirit of a similar question for the harmonic series, is there a way to regularize the (divergent) sum of all primes? $$ \sum_{p \text{ prime}} p $$ Neither of these questions obtained a ...
user76284's user avatar
  • 1,793
1 vote
2 answers
141 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\...
Sylvain JULIEN's user avatar
2 votes
1 answer
525 views

Sets of primes with a given Frobenius conjugacy class

Let $a$ and $b$ be two relatively prime positive integers. Denote by $S$ the set of all positive primes of the form $a+bn$ (where $n$ is an integer, possibly zero or negative). Let $S'$ be any set of ...
user avatar
2 votes
0 answers
156 views

Questions about a certain sequence of naturals generated by primorials

I'm working on the following sequence of naturals (which is NOT listed in OEIS) $$3,5,11,17,23,29,59,89,119,149,179,209,419,629,839,1049,1259,1469,1679,...$$ whose elements are generated this way $$3=(...
Augusto Santi's user avatar
2 votes
0 answers
119 views

On the set $\{n>0:\ n\ \text{is a quadratic nonresidue modulo the}\ n\text{th prime}\}$

Let $S$ denote the set of positive integers $n$ with $n$ a quadratic nonresidue modulo the $n$th prime $p_n$. The first 20 elements of $S$ are $$2,\, 3,\, 6,\, 7,\, 8,\, 10,\, 11,\, 13,\, 15,\, 18,\, ...
Zhi-Wei Sun's user avatar
  • 14.4k
0 votes
1 answer
352 views

A sufficient condition for a set of primes to be the set of reducibility of an integer polynomial

Let $P$ be the set of all positive primes. Let $S$ an arbitrary infinite subset of $P$ satisfying the following assumption: there exists a finite Galois extension $K$ of $\mathbb{Q}$ and a conjugacy ...
user avatar
0 votes
1 answer
132 views

A density zero set of primes dividing the values of a non-constant integer polynomial

For a given $P\in \mathbb{Z}[x]$ call a positive prime $p$ good if there exists $n\in \mathbb{Z}$ such that $p$ divides $P(n)$. Does there exist a non-constant $P$ such that the set of good primes has ...
user avatar
5 votes
1 answer
164 views

Representation of primes of the form $4m+3$ with double radicals

Let $\,q\,$ be a prime of the form $\,4\, m_q+3$. I ask if it is always possible to find two primes $\,p_1$ and $\,p_2$ of the form $\,4\, m_p+1$ such that $$q=\sqrt{p_1+\sqrt{p_2+q}}$$ E.g. $$3=\...
Augusto Santi's user avatar
2 votes
1 answer
289 views

Convergence of series $\sum_{k=1}^{\infty}\frac{p_{k+1}-p_k}{(p_{k+1}+p_k)^\alpha}$

I ask if the series $$\sum_{k=1}^{\infty}\frac{p_{k+1}-p_k}{(p_{k+1}+p_k)^\alpha}$$ where $p_k$ stands for the prime of index $k$, has the same properties of convergence of the series $$\sum_{k=1}^{\...
Augusto Santi's user avatar
-1 votes
1 answer
242 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 ...
user142929's user avatar
-3 votes
1 answer
178 views

Asymptotic behavior of $\sum_{k=1}^{n}\frac{p_{k+1}}{p_{k+1}-p_k}$

I refer to my previous question Asymptotic behavior of a certain sum of ratios of consecutives primes. We can split the sum $$\sum_{k=1}^{n}\frac{p_{k+1}+p_k}{p_{k+1}-p_k}$$ where $p_k$ stands for the ...
Augusto Santi's user avatar
10 votes
1 answer
465 views

Asymptotic behavior of a certain sum of ratios of consecutives primes

I am looking for the asymptotic growth of the following sum $$\sum_{k=1}^{n}\frac{p_{k+1}+p_k}{p_{k+1}-p_k}$$ where $p_k$ stands for the prime of index $k$. Manual computations show, for small values ...
Augusto Santi's user avatar
6 votes
3 answers
821 views

Is it possible to multiply two series to get as a result all composite numbers?

I was toying with the following problem: Is it possible to find two infinite integer sequences $(a_n), (b_n)>0$ such that $\sum_{n=1}^{\infty}\frac{1}{(a_n)^s}\cdot \sum_{n=1}^{\infty}\frac{1}{(b_n)...
Konstantinos Gaitanas's user avatar
6 votes
0 answers
197 views

Smooth integers with lower bound on $\omega(n)$

Define $(b,c)$-smooth integers to be integers having all prime factors bigger than $c$ and smaller than $b$. Probability a number is $(b,1)$-smooth is governed by the Dickman function while ...
VS.'s user avatar
  • 1,816
9 votes
1 answer
415 views

Conjectured primality test for specific class of $N=4kp^n+1$

Can you provide a proof or counterexample for the following claim? Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ . Let $N= 4kp^{n}+1 $ where $k$ is a positive natural number , $...
Pedja's user avatar
  • 2,673

1
11 12
13
14 15
41