Prime numbers are natural numbers greater than 1 not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

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Trying to understand how Lehmer's method represents a simplification of Meissel's method for counting primes

My question stems from a wikipedia article on prime counting. The details on Meissel's method can be found in the wikipedia article. As I understand, Meissel proposed two formulas which I asked ...
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The nth roots of $z_n=p_n\cdot(i)^{p_n}$, where $i=\sqrt{-1}$ and $p_n$ is the nth prime number

I want refresh some basics too in Complex Analysis. Let $p_n$ the sequence of prime numbers $2, 3, 5, 7\ldots$, thus $p_n$ is the general term of this sequence, and $i=\sqrt{-1}$ is the complex ...
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prove this inequality for sufficiently large n

Prove the following inequality for $n \geq 100$ $$\pi\bigg(\frac{\log (n^2+2n)}{\log 2}\bigg) < \pi(2n) - 1.5\pi(n)$$ $\pi(n)$ denotes to prime counting function
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Pseudoprime $n$ to the base $b$ relation to $\prod\limits_{p\mid n}\gcd(p−1, n−1)$

Let $n > 1$ be an odd number, and let $P_n = \{b\pmod{n} : b^{n−1}\equiv 1\pmod{n}\}$ be the set of bases $b$ for which $n$ is a pseudoprime to the base $b$. Prove that $|P_n| = ...
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Linear convex combinations of $Li(x)=\int_2^x\frac{1}{\log(t)}dt$ and $\frac{x}{\log(x)}$, and prime counting function

Can provide us a linear convex combination of $Li(x)=\int_2^x\frac{1}{\log(t)}dt$ and $\frac{x}{\log(x)}$ a better approximation for $\pi(x)$, the prime counting function? Or not, is better $Li(x)$ ...
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Do we have more primes of the form $3k+1$ or of the form $3k+2$?

Let us denote by $a_{1}(n)$ the number of prime numbers in the set $\{1,2,...,n\}$ which are of the form $3k+1$. Let us denote by $a_2(n)$ the number of prime numbers in the set $\{1,2,...,n\}$ which ...
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Bounds on twin prime counting function

I read somewhere (unfortunately I cannot find the paper again) that the twin prime counting function $\pi_2(x)$ satisfies $\pi_2(x) \leq C\frac{x}{\log^2x}$ for some constant $C$. How would one prove ...
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Can it be proven that infinite many primes can be formed only using two distinct digits?

It seems obvious that infinite many primes can be formed only using two distinct digits. $67776767776667777777$ is an example for such a prime. Even if we allow only the digits $0$ and $1$, there ...
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Zeros of the prime zeta function

A basic confusion about zeros of the prime zeta function $P(s).$ Let $s= \sigma+i~t$ with $\sigma>0.$ Letting $C(s)$ be the corresponding composite zeta function we have ...
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A binary irrational with bits defined by primes

Define a number $q$ in binary notation whose $n$-th bit is $1$ for $n$ prime, and $0$ for $n$ composite. So its 2nd, 3rd, 5th, 7th, 11th, etc. bits are $1$, with all other bits $0$. Here is $q$ out to ...
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Squares of $F_{467}$ and 7 as a quadratic residue of a prime mod that prime

I am having a hard time understanding what exactly is meant by this question. Could someone give me a solution with a clear explanation? If $x=467$, are 111 127 and 225 squares in $F_x$? Explain your ...
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primes of the forms $…54321012345…$ and $…543212345…$

I'm looking for primes formed from concatenating the first n natural numbers in these following manners: (n)(n-1)(n-2)$...54321012345...$(n-2)(n-1)(n) (n)(n-1)(n-2)$...543212345...$(n-2)(n-1)(n) ...
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Carmichael number $p \times … \times q$ , such there are at most $20$ primes in the range $[p,q]$

Let $N$ be a Carmichael number , $p$ its smallest prime factor, $q$ its largest prime factor. Which is the largest possible prime $p$, if the range $[p,q]$ only contains at most $20$ primes ? The ...
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For which $k$ do we get a Carmichael-number?

Let $y$ be the vector ...
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Constructing a number strong probable prime to several bases

See here https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test for the definition of a strong probable prime, where it is also mentioned that Arnault constructed a number being strong ...
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The last p digits of $p^p$ form a prime number,where p is a prime

Let p be a prime number, here I'm interested with the last p digits of $p^p$ if it forms a prime number. And if I've not mistaken, the smallest p with that property is $433$. Meaning that the last ...
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Greatest Common Divisors in columns and rows of Pascal Triangle

Let $n$ and $k$ be integers such that $n\ge3$ and $k\ge 2$ and $g(n)$ is the prime gap where $n$ lies $$k\le g(n)+2\implies \gcd\left(\binom{n+j}{k} , j\in \{ 1,k-1 \} \right)\gt1$$ ...
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Polynomials producing Carmichael numbers

It is well known that $$(6n+1)(12n+1)(18n+1)$$ is a Carmichael number, if all factors are prime. I found the polynomials $$(6n+1)(12n+1)(18n+1)(36n+1)$$ $$(18n+1)(36n+1)(108n+1)(162n+1)$$ ...
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Asymptotic local limit theorem and applications in analytic number theory

I'm wondering if one could get similar results to the classical local limit theorem if one assumes that conditions, such as independence and identicallity of distribution of the random variables ...
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Representing integers as difference of semi-primes

Can every integer, $k$ (where $4 \le k < n$), be written as the difference of two semi-primes whose prime factors are at most $n$ ? or put another way: Given positive integers $k\ge 4$, $n > ...
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Some questions about $d$-primes

Here : https://oeis.org/search?q=3%2C7%2C13%2C31%2C103%2C109%2C151&sort=&language=&go=Search the primes are mentioned which remain prime if the digit $1$ is inserted at any position, ...
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Power series with products of prime numbers as coefficients

I encountered in my work an infinite series (only even power) with $n^{th}$ coefficient being of the following form: $$\frac{1}{3\cdot5\cdot7\cdot11\cdot...\cdot(n^{th} prime)}.$$ I wonder if such ...
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Which odd composite numbers $n$ are strong-pseudoprime to no base $a$ with $1<a<n-1$?

In this question : A composite odd number, not being a power of $3$, is a fermat-pseudoprime to some base I did not hit my own intend. I only asked for the numbers $n$ that are not a ...
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Are minus-primorial-primes rarer than plus-primorial primes?

Here https://primes.utm.edu/glossary/page.php?sort=PrimorialPrime it can be seen that the largest known primorial prime of the form $p\#-1$ is far smaller than the largest known primorial prime of ...
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Is $49931$ the least positive integer $k$, such that $312^{4009}+k$ is prime?

Denote $$z(k)\ :=\ 312^{4009}+k$$ After a long search, I found the (probable) prime $z(49931)$. Questions : $1)$ Is $k=49931$ the least positive integer, such that $z(k)$ is prime ? ...
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How to calculate the $n$ prime from $\pi (n)$?

Assume we had an exact formula for $\pi (n)$, how could we get from that formula an exact expression for the $n$th prime? I tried looking at approximations we have of $\pi (n)$ like $\frac {n}{\ln ...
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On n! divided by a product of primes and related questions

We have the following Definition 1. For integers $n\geq 1$ we define $$f(n) = \begin{cases} 1, & \text{if $n=1$} \\[2ex] \frac{n!}{\prod_{p\leq n}p}, & \text{if $n>1$} ...
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Finding primes using the Fibonacci sequence in modular form

I was wondering if the following is already a known result in mathematics. I have tested it and it seems to work every single time. If I write the Fibonacci sequence in $\bmod (a)$ form and it ...
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Prime from mirror concatenation of first primes

The mirror concatenation of the first 1, 6 and 8 prime numbers with no primes being reversed is a prime ! i.e. 131175323571113 and 19171311753235711131719 are prime numbers! (beautiful primes!). After ...
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Greatest prime factor of $\left(\dfrac{n(n+1)}{2}\right)^2-1$.

Consider $$ \left(\dfrac{n(n+1)}{2}\right)^2-1. $$ Is is possible to say something about the lower bound on the greatest prime divisor of the above expression depending only on $n$? I surfed through ...
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Generalization of a Result Concerning Projective Planes

Let $\mathcal P$ denote the set of all possible orders of projective planes. For $q\in\mathcal P$, let $PG_2(q)$ denote the projective plane of order $q$. There is a theorem due to James Singler ...
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Does the Euler product for the Dirichlet $\beta$-function converge for all $\Re(s)>\frac12$?

The Dirichlet $\beta$-function is defined for $\Re(s)>0$ as: $$\beta(s) = \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^s}$$ It has the following Euler product (I used that Dirichlet character ...
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How many entries in the sequence $x_n$ given by recursion $x_1=1\ ,\ x_{n+1}=p_{x_n}$ are known?

The sequence $x_n$ with the recursion $x_1=1\ ,\ x_{n+1}=p_{x_n}$ , where $p_k$ denotes the $k-th$ prime, has the following values : ...
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If $q$ is a prime, $gcd(x(x+2),q\#)=1$ and $q < x < q^2$, doesn't it follow that $x,x+2$ are twin primes?

I recently asked a question that was not well received. That's ok. I don't disagree with the ratings if my question is unclear. I want to verify the foundation of my reasoning. Doesn't it follow ...
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Prove an inequality of composite numbers

Yesterday after reading this post I tried to prove the inequality as given in the post. The inequality is, $$c_m+c_n>c_{m+n}$$ for all $m,n\ge1$. The problem was regarding the following special ...
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A multivalued function $ f(x) = 0 $ with integer solutions $ x_1=p(n) $ and $x_2=q(n) $

Please help me to define a multivalued function $ f(x) = 0 $ with integer solutions : $ x_1 = p(n)$ and $ x_2 = q(n) $ such that $\dfrac{ p(n) + q(n) } { 2 } = 2 n + 1 $ and $ ...
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lattice walks with primes and composites

In the regular square lattice create a walk moving according the value of a counter. Consider two types of walks: In the first walk advance forward one unit if the counter is a composite number and ...
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The name given to the number 1 in the context of Primes and Composites

We give names to the sets of numbers called Primes and Composites. Is there a name for the number 1, in this context, seeing it is neither a Prime or Composite?
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Prime distribution around Riemann counting function

Is it true that the primes are normally distributed around the Riemann counting function $R(n)$ as a folded CDF? (Scaled $p_n-R(n)$)
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Question about the Chinese Remainder Theorem and the residue class ring ${\bf Z}/p\# {\bf Z}$

In a question that I asked on MO, Terence Tao observed that: The Chinese Remainder Theorem tells us that the residue class ring ${\bf Z}/p\# {\bf Z}$ is isomorphic (as a ring) to the product of ...
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Numbers $717, 71717, 7171717,\dots$ and primality

Prove or disprove that all numbers $717, 71717, 7171717,\dots$ are composite. This is related to this question. $\begin{array}\\ 717 &= \text{div by 3}\\ \color{blue}{71717} &= 29\cdot ...
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Solution to a set of number theoretic constraints

I'm trying to prove a graph theoretic lemma for my research; I need to construct graph homomorphisms between some delicately defined graphs. I believe I can do this if (and maybe only if) I can find ...
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Is there a strong version of twin prime conjecture?

The twin prime conjecture states that : There exists infinitely many integers such that $n$ and $n+2$ are both primes for a fixed $k$, can we find integers $a_1,a_2,\cdots, a_k$ such that: ...
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Primality Test for $N=2\cdot 3^n-1$

Definition Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$ , where $m$ and $x$ are nonnegative integers . Conjecture Let $N=2\cdot 3^n-1$ such ...
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Count of lower and upper primitive roots of prime $p \equiv 3 \bmod 4$

I was exploring the layout of primitive roots of primes over a reasonable range and this question concerns the number of primitive roots either side of $p/2$. Many primes have an exact match between ...
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Convergence of recurrence relation involving divisors

I$\let\leq\leqslant\let\geq\geqslant$ thought up a family of sequences, recursively defined by $$a_{n+1}=\frac{d_n^ra_n+a_{d_n}}{d_n^r+1}\quad(n\geq2)$$ where $r,a_1,a_2\in\mathbb R$ are parameters ...
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Numbers $m,n$, such that $a^m+ab+b^n$ is always composite

Are there integers $m,n\ge 2$, such that $$a^m+ab+b^n$$ is composite for all integers $a,b\ge 2$ ? I checked the pairs with $2\le m\le 100$ and $2\le n\le 100$ and always found a prime of the form ...
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Multiplication Sieving

Background I made another improvement to Fermat's sieve factorization, by merging the sieves groups of $4,3,5,7,11,13,17$ in two one big group. This method allows me to reduce the values that I need ...
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Proof of an inequality about primes

I'm very new to number theory and looking for a proof of the following inequality: $$c' \log^{\text{#} \mathbb{P}}{R} \leq \sum \limits_{\substack{n \leq R\\p|n \implies p \in \Bbb P}} 1 \leq c ...
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Is an upper bound known for the least exponent $k$ such that $(\pi(n^k)-\pi(n))_{n=1}^\infty$ is strictly increasing?

Is an upper bound known for the least exponent $k$ such that $(\pi(n^k)-\pi(n))_{n=1}^\infty$ is strictly increasing? It appears that the sequence is strictly increasing when $k\geq2$, but ...