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.

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0
votes
1answer
88 views

Analytic continuation and convergence of a Riemann zeta related function

The functions in question are $$L(s)=\sum_{k=1}^\infty \frac{\lambda(k)}{k^s}=\frac{\zeta(2s)}{\zeta(s)} \mbox{ and } L^*(s)=\frac{1}{2}\sum_{k=1}^\infty \frac{\lambda(k)+(-1)^{k+1}}{k^s}=\frac{L(s)+\...
9
votes
1answer
161 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 ...
1
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0answers
150 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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0answers
33 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 ...
0
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0answers
55 views

On complexity of a particular prime problem

Is the following problem in $PH$ and is it complete for any class? Problem: Is the $i$th bit of the $m$th prime $1$? It appears to require a counting quantifier which has to demonstrate witness is the ...
1
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1answer
67 views

Cyclotomic Numbers, Difference Sets

I have reading papers by Cunsheng Ding on Binary Cyclotomic Generators, Linear Complexity of Generalised Binary Sequences of Order 2. Since the topic is new to me understanding the text is quite ...
2
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1answer
139 views

A family of difference sets (paper by A. L. Whiteman)

I was reading A. L. Whiteman - A family of difference sets. On page 109, the author generalizes the reduced residue system modulo $v$ where $v=pq$, $p$ and $q$ are primes. At the last paragraph, can ...
-6
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0answers
124 views

Sieve methods and primal height

Define the primal order $\omega_{\mathbb{P}}(n)$ of an integer $n$ as the smallest $k$ such that the $k$-th iterate of the prime counting function $\pi$ evaluated at $n$ is composite and the primal ...
10
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1answer
821 views

Normal numbers, Liouville function, and the Riemann Hypothesis

This is a question about whether or not some number $\lambda^*$ is normal in base 2. More specifically, I am wondering if $\lambda^*$ is not normal. Proving it is normal would be next to impossible, ...
2
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0answers
104 views

On the connection between sums of prime numbers and distribution of prime numbers

As an amateur mathematician, I have always been fascinated by the magic of prime numbers, and their apparently random distribution. I was utterly amazed when I found the following connection between ...
2
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1answer
136 views

Truncated Euler products, Dirichlet eta function, and convergence issues

Can you prove that the following series does not converge if $\frac{1}{2}<\sigma<1$, no matter how close to $1$ sigma is, and no matter how large $t>0$ is? The series is defined as $$W(\sigma,...
3
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1answer
201 views

Formulas for $\sum_{x=1}^{n}\Big\{\frac{x^q}{n}\Big\}$

Consider sum: \begin{equation} S_q(n) = \sum_{x=1}^{n}\Big\{ \frac{x^q}{n} \Big\} \end{equation} where $\{x\}$ is fractional part of $x$. It's easy to see that $S_{1}(n) = \frac{1}{2}(n-1)$, but ...
-1
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0answers
95 views

Gap repeating integers

Under Goldbach's conjecture, set $r_{0}(n):=\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$ and $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$. Define similarly $r_{i+1}(n):=\inf\{r>r_{i}(n),(n-r,n+r)\in\...
-1
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1answer
181 views

A question on assigning finite values to divergent sums involving expression of primes

We know the following: $$\gamma=\lim_{n\to\infty }\left(\sum_{k=1}^n\frac{1}{k}-\ln(n)\right).$$ This could be a good candidate for renormalized sum of $\left(\sum_{k=1}^{\infty}\frac{1}{k}\right)$. ...
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0answers
152 views

Does one have $\alpha_{n}\ll\sum_{p^m\leq n, m\geq 2}\Lambda(p^{m})\log n$?

This question is a follow-up to my question "About Goldbach's conjecture" (direct link: About Goldbach's conjecture) whose beginning I copy-paste below: "Let's consider a composite ...
5
votes
1answer
339 views

Weak Goldbach conjecture with distinct primes for odd integers between $4\times 10^{18}$ and $10^{27}$

This is related to the conjecture that all odd integers greater than $17$ can be written as the sum of 3 distinct primes. Schinzel showed that the Goldbach conjecture implied this in 1959 and as the ...
1
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2answers
302 views

Twin Primes- Clement conjecture proof

For a work I'm doing, I need to provide the proof of the Clement theorem $$ \text{($n$ and $n+2$ are twin primes)} \quad\Longleftrightarrow\quad 4[(n-1)!+1]+n \equiv 0 \pmod{n(n+2)}. $$ So I decided ...
1
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1answer
138 views

What is a non-trivial upper bound on the $k$th prime below a given prime $p$?

Given a prime number $p_0$, by Bertrand's postulate we know that \begin{gather} p_1\ge\frac{p_0}{2}\\ p_2\ge\frac{p_1}{2}\ge\frac{p_0}{2^2}\\ \vdots\\ p_k\ge\frac{p_0}{2^k} \end{gather} where $p_1,p_2,...
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0answers
118 views

Integers having the same digits as their prime factors

Say an integer is "digitally conservative" in base $b$ if the set of its digits coincides with the set of the digits of its prime factors, like $37127=137\times 271$ in base $10$, and denote ...
-2
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1answer
128 views

Divisors of numbers [closed]

Prove for any given (n,m) there are n consecutive numbers that m divides the number of their divisors. I know that $p_1^{\alpha_1}p_2^{\alpha_2} ... p_n^{\alpha_n}$ have $(\alpha_1+1)(\alpha_2+1)... (...
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0answers
121 views

Primality radius and Chebyshev's bias

Under Goldbach's conjecture, denote as usual by $r_{0}(n)$ the smallest positive integer $r$ such that both $n-r$ and $n+r$ are prime for $n$ a large enough composite integer. Obviously $p_{0}(n):=n-...
1
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0answers
74 views

Approximating real numbers [closed]

Given any two different prime numbers $p_1$ and $p_2$, any real number $r$ and an $\epsilon$. Can you always find two integers $n$ and $m$ such that $|\frac{p_1^n}{p_2^m} -r| \lt \epsilon)$? Here is a ...
4
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1answer
241 views

Is $\prod_{k=1}^nk^{\sigma(k)}$ a square or a cube for some $\sigma\in S_n$?

Note that for any permutation $\sigma\in S_5$ the product $\prod_{k=1}^5k^{\sigma(k)}$ is neither a square nor a cube. Question. Let $n>5$ be an integer. Is the product $\prod_{k=1}^nk^{\sigma(k)}$ ...
2
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0answers
112 views

Primes of the form $\frac{n^2-n+4}{2}$ satisfy Hardy-Littlewood analogue?

Let $n,a,b$ be positive integers with $a<b$. Consider primes of the form $f(n)=\dfrac{n^2-n+4}{2}$. Let $C(a,b)$ denote the amount of primes of the form $f(n)$ between (and including) $f(a)$ and $f(...
1
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0answers
80 views

Smooth number pairs satisfying a congruence

Let $\mathcal P=\{p_1,\dots,p_{2t}\}$ be $2t$ primes between $2^\ell$ and $2^{\ell+1}$ and fix an exponent bound $a\in\mathbb Z_{\geq2}$. Fix $N\in\mathbb N$ whose prime factors $p$ satisfy $p>2^{\...
4
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1answer
209 views

The Diophantine equation $ (xz+1)(yz+1)=az^{3}+1$ has no solutions in positive integers with $z > a^2+2a$

Problem. Let $a$ be a positive integer that is not a perfect cube. Show that the Diophantine equation $(xz+1)(yz+1)=az^{3}+1$ has no solutions in positive integers $x, y, z$ with $z > a^{2}+2a$. ...
0
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0answers
53 views

Primality of $n$ bit integers in depth $n^\alpha$ under standard conjectures?

Denote $\mathsf{NC}(\mathsf{SUBLINDEPTH}(n),f(n))$ to be set of boolean circuits of fan-in $2$ which can be represented by depth $\cap_{\alpha>0}\mathsf{}n^\alpha$ and $f(n)$ sized Boolean circuits....
9
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0answers
214 views

Is $\pi (x)=\operatorname{R}(x)-\sum_{\rho}\operatorname{R}(x^{\rho})$ correct at all?

It should be the case that, in some appropriate sense $$\pi (x)\sim \operatorname{Ri}(x)-\sum_{\rho}\operatorname{Ri}(x^{\rho}) \tag*{(4)}$$ with $\operatorname{Ri}$ denoting the Riemann function ...
0
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0answers
190 views

Are there infinitely many solution $(a,b,p)$ satisfying $a^b+b^a=p$?

Let $a,b$ be integers which satisfy $2\leq a\leq b$ and $p$ a prime number. Are there infinitely many solution $(a,b,p)$ satisfying the following equation? \begin{eqnarray} a^b+b^a=p \end{eqnarray} I ...
0
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1answer
123 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 $...
-1
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1answer
108 views

Prove using Dyck naturals: for $n \in \mathbb{N}_{+}$ and big enough $k \in \mathbb{N}_{+}$, $p_{k-1} < \cdots < np_{k-a_{n}}$ (a is A073093)

While conducting research in connection with arXiv:2102.02777 ("Recursive Prime Factorizations: Dyck Words as Numbers"), I noticed certain interesting patterns, one of which inspired the ...
13
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2answers
2k 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 ...
21
votes
1answer
716 views

How to see that the determinant of this matrix is nonzero for all primes?

I'm trying to show that $\sum_{i = 0}^{p-2} (i+1)^{-1} t^{i+n}$ where $0 \leq n \leq p-2$ spans the vector space $\mathbb{F}_p[t]/(1-t)^{p-1}$ as a rank $p-1$ module over $\mathbb{F}_p$. In other ...
2
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0answers
76 views

On a result of Euler on pseudoprimes

In several sources (for instance on page 58 of the first ed. of Crandall & Pomerance book on prime numbers or at the end of this paper by J. H. Jaroma), I have seen a result that goes like this: ...
0
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0answers
85 views

$f$ - arithmetic functions

Let $\mathbb{P}$ denote the set of primes. Let $f$ be a real-valued function such that it satisfies the following: $f(p) > 0$ for primes $p$, $\lim_{p \to \infty} f(p) = \infty$, and $f(p) \leq p$ ...
6
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1answer
409 views

Set of prime numbers $q$ such that $\sum\limits_{p\leq\sqrt{q}}p=\pi(q)$, where $p$ are prime numbers

The question is: does the set of prime numbers $q$ such that $\sum\limits_{p\leq\sqrt{q}}p=\pi(q)$, where $p$ are prime numbers, contain infinitely many elements? You can find the first elements here (...
0
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0answers
77 views

Primes in many variables polynomials form

As in this question >> Primes of the form $x^2 + y^2 + 1$ There are $\asymp \dfrac{x}{\log^{3/2} x}$ primes of the form $a^2+b^2+1$ up to $x$. I read a book stated that there exist a polynomial $...
5
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0answers
133 views

Small multiplicative order modulo infinitely many primes

Let $a>1$ be an integer. Artin's conjecture claims that if $a$ is not a perfect square, then $a$ is a primitive root modulo infinitely many primes $p$ (which moreover form a subset of positive ...
4
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0answers
175 views

On $\sum_{n=1}^\infty\frac1{p_1+\cdots+p_n}$ and $\sum_{n=1}^\infty\frac1{p_1\cdots p_n}$

For $n=1,2,3,\ldots$ let $p_n$ denote the $n$-th prime number. Question. Are the two numbers $$c_1=\sum_{n=1}^\infty\frac1{p_1+\cdots+p_n}\ \ \ \ \text{and}\ \ \ \ c_2=\sum_{n=1}^\infty\frac1{p_1\...
5
votes
1answer
198 views

Quadratic Diophantine equations with all values prime

Given a quadratic Diophantine equation over the integers in two variables, can we say much about when it has only finitely many solutions with the additional assumption that both variables are prime? ...
0
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0answers
56 views

Gaussian primality radius

I'm trying to generalize the notion of primality radius of a positive composite integer $n$, defined as a positive integer $r$ such that both $n-r$ and $n+r$ are prime, to Gaussian integers. As such, ...
1
vote
1answer
110 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 ...
1
vote
0answers
104 views

Numbers whose digits, in order, display prime factors

There's a post in CodeGolf which asks for code to find numbers whose digits contain their prime factors without rearrangement. The author suggests the mathematical definition is "Determine if ...
15
votes
2answers
695 views

Is the surreal number $\omega(\sqrt{2}+1)+1$ a prime?

In the 1986 book An Introduction to the Theory of Surreal Numbers, Gonshor, on page 117, notes that it is an open problem whether $\omega(\sqrt{2}+1)+1$ is a prime, using the standard definition of ...
8
votes
6answers
646 views

How many solutions are there to the equation $x^2 + 3y^2 \equiv 1 \pmod{p}$?

Let $p$ be a prime. How many solutions $(x, y)$ are there to the equation $x^2 + 3y^2 \equiv 1 \pmod{p}$? Here $x, y \in \{0, 1, \ldots p-1\}$. This paper (https://arxiv.org/abs/1404.4214) seems like ...
0
votes
1answer
205 views

How differently would we model the distribution of primes if prime gap is larger?

Cramer's conjecture based on his random model provides prime gaps are bound by $O(\log^2p_n)$ where the gap is between $(n+1)$th and $n$th prime. How differently would primes be modeled if gaps of $O(...
0
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0answers
77 views

On a generalised result of Mertens

Let $\varphi$ be the Euler totient function, $N_k$ denote the product of the first $k$ primes and define $$f(k, r) = \frac{(N_k)^r}{\varphi((N_k)^r) \log \log ((N_k)^r)}.$$ where $r \in \mathbb{N}$. ...
50
votes
5answers
7k views

What is the simplest proof that the density of primes goes to zero?

By density of primes, I mean the proportion of integers between $1$ and $x$ which are prime. The prime number theorem says that this is asymptotically $1/\log(x)$. I want something much weaker, namely ...
0
votes
1answer
61 views

What are the complexity classes of these problems about divisibility and coprimality?

The problems 'Given $0<a<b$ and a prime $p<a$ is there an integer $\ell\in[a,b]$ such that $p|\ell$?' 'Given $0<a<b$ and an integer $q\not\in[a,b]$ is there an integer $\ell\in[a,b]$ ...
4
votes
0answers
191 views

What is the complexity class of this problem without Cramer's conjecture?

The problem 'Given $0<a<b$ is there a prime in the interval $[a,b]$?' is in $\mathsf{NP}$. If we assume Cramer's conjecture the problem is in $\mathsf{P}$ since if $b-a>(\log a)^{2+\epsilon}$ ...

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