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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Number of primes of a certain form

Let $p_n$ be the nth prime. Are there an infinite number of primes of the form $2p_n+1$? Is something known about questions like this?
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Showing the infinitude of primes using the natural logarithm

I came across this proof in Proofs From the Book by Aigner and Ziegler. It uses the inequality $logx \leq \pi(x)+1$. (Here, we use natural logarithm) The proof starts with the inequality $log$ $x \...
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Summing the digits of sum of the digits to obtain prime numbers

Define sum of digits (in base $10$) function as $sd_{10}(n)=\sum_{i=0}^ma_i$ where $n=\sum_{i=0}^ma_i \cdot 10^i$ and $0\le a_i\le 9$. If we choose prime number $86423$ and sum its digits we obtain a ...
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simplifying a sum with modular arithmetic

Let $p\!\geq\!3$ be a prime and $n\!\in\!\mathbb{N}$. For $i\!=\!1,\ldots,n$ let $w_i\!=\!2i\!-\!n\!-\!1$. Let $n\%p$ denote the remainder in the integer division of $n$ by $p$. Can the following sum ...
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Is there anyway to find how many prime factors has a composite number without knowing them?

Let's call f(n) the function that gives us the number of different prime factors of a composite number n For example: f(24)=2 Let's call g(n) the function that gives us the number of prime factors of ...
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Lehmann primality test

How to calculate final probability that a given number is prime after 1000 iterations, when using Lehmann primality test ?
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A very nice pattern involving prime factorization

A while ago I was fiddling around with prime numbers and C++. I defined: $$f_a(b)= \text{ the amount of numbers } 2^a\leq n<2^{a+1}\text{ with } b \text{ prime factors}$$ I calculated $f_a(b)$ for ...
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Prove, by giving an example , Fermat's Little Theorem

Prove, by giving an example, that, if n is not prime, a≠0(mod n) then it is not necessarily true that { [1]n,[2]n.........[n-1]n} = {[a.1]n,[a.2]n,.......[a.(n-1)]n} could you give me any hint to ...
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A necklace problem (related to the modulo function)

Suppose we have (non-closed) necklace with a large amount of white beads on it, and we wish to colour those beads to make a nice pattern. Unfortunately we are kind of picky about the colours we will ...
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Erdos' Arithmetic Progression of Polignac's Numbers

Paul Erdos has proven that there is an infinite arithmetic progression of Polignac's numbers - odd numbers that cannot be represented as a sum of a prime and a power of $2$, in Erdos, Paul. "On ...
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Distribution of final digits of consecutive primes

There's been a lot in the press recently about the unexpected distribution of final digits in pairs of consecutive primes, and many people have written programs to confirm the observation that pairs ...
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Looking for help on writing a mathematical argument clearly and concisely

Let $p_n$ be the $n$th prime and $a < p_n$ be a non-negative integer . Let $f(a,p_n)$ be the number of integers $x$ such that: $$a(p_{n-1}\#) < x < (a+1)(p_{n-1}\#) \text{ and } \...
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Would this be an easier way to prove the twin prime conjecture?

Prove: For every prime, $p\geq7$, there exists some $pn$ such that $p$ is its largest prime factor, $n$ is a positive integer, and $(pn-4, pn-2)$ is a twin prime. My questions: Would this indeed ...
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A Conjecture Sharper than Cramér's and Firoozbakht's

Notation: $\lfloor\cdot\rfloor$ is floor function; and $\pi(x)$ is the prime-counting function up to $x$. $g_k := p_{k+1} - p_k$ . OEIS sequence A267549 is "Primes prime(k) such that floor( (prime(k)/...
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Is the following claim true about systems of quadratic congruences modulo consecutive prime numbers

Is the following true? Choose any value for $y : y \in \mathbb{N}$ If $N(y)$ is the smallest natural number that satisfies the following system of quadratic congruences: $N(y)^2 \not\equiv 1$ ...
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Given a number, show that there exists another number obeying certain congruence relations relative to the first

Given an even number $2x$, show that if: $2x \equiv a_1 \bmod p_1; 2x \equiv a_2 $ mod $p_2 ; \ldots ; 2x \equiv a_y$ mod $p_y$ (where $p_1,p_2,\ldots,p_y$ are consecutive prime numbers strictly ...
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Is there a short proof of the existence of $a$ so that $a$ is a primitive root for infinitely many primes $p$?

After looking for a general answer I found Artins conjecture, and I was happy to see so much is known. However I don't know nearly enough to follow the proof, yet it bothers me I can't prove the ...
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Finite amount of consecutive smooth numbers

is there a short proof of the fact that there is a finite amount of consecutive smooth numbers (meaning Given a finite set B, there is a finite amount of pairs $n,n+1$ so that both can be expressed as ...
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Number-theoretic asymptotic looks false but is true?

Question Let $p_r$ be the $r'th$ prime. Is it true that, $$\sum_{r=1}^\infty s^r \ln(p_r) \sim \frac{s}{(1-s)} $$ I know this looks bizarre but kindly consider the argument below. I'm also ...
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Quantitative aspect of Chebotarev Density Theorem

I recently learned the Chebotarev Density Theorem for global fields. As far as I have seen, all applications of CDT seem to focused around proving some set of prime ideals (or places in function ...
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Proof that the spectrum of prime distribution will give zeros of Riemann Zeta function

All: Many of us have read that the spectrum of prime distribution will give zeros of Riemann Zeta function. For example, Mazur and Stein's book: (http://wstein.org/rh/rh.pdf ) have many nice pictures ...
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Paper of Paul Erdös

I'm trying to understand On Arithmetical Properties of Lambert Series by Erdös, but am stuck on the first page. He states: Put $k=\left[(\log n)^{1/10}\right]$ and let $p_1,p_2,\ldots$ be the ...
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Modular Euler product?

We know the Euler product. $$\zeta (s)=\prod_{p}\frac{p^{s}}{p^{s}-1}$$ I wonder if there is formula or any kind of work for this kind of prime product below? $$\prod_{p\equiv a \ (mod \ b)}\frac{p^...
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How to show this identity is true?

$$\left(\sum_{k=1}^{\infty}\frac{i^{\Omega (k)}}{k^{s}}\right)^{2}=\frac{\zeta (4s)}{\zeta (2s)}\frac{2^{s-1}}{i-2^{s-1}}$$ where $i=\sqrt{-1}$ and $\Omega (k)$ is the number of (not necessarily ...
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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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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| = \prod\limits_{p\...
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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 $$(1)\hspace{15mm}P(s)+C(...
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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) I'...
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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 $$k\le g(n)+2\implies \gcd\left(\binom{n+j}{k} , j\in \{ 1,k-1 \} \right)\gt1$$ $\binom{n+j}{...
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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 $$(6n+1)(12n+1)(18n+1)(36n+1)$$ $$(18n+1)(36n+1)(108n+1)(162n+1)$$ $$(20n+1)(...
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Can every odd prime $p\ne 11$ be the smallest prime factor of a carmichael-number with $3$ prime factors?

According to my search, the number $561=3\times11\times17$ is the only carmichael-number with $3$ prime factors, which is divisible by $11$. Is this true ? If yes, $11$ cannot be the smallest ...
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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 $k\ge 4$, $n > k$...
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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 fermat-...
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Techniques in analytic number theory

I'm fairly new to the subject and trying to figure things out. Would be nice to hear some ideas and trickery for what follows. Suppose we wish to show there exists an integer $x$ in some finite set $...
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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 ...