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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Examples of Weil's explicit formula

In Bombieri, PROBLEMS OF THE MILLENNIUM: THE RIEMANN HYPOTHESIS, Clay Mathematics Institute (2000), from page 8, V. Further evidence: the explicit formula the author tell us that there is a ...
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Dirichlet inverse of $(-1)^n$

I was tinkering around and noticed the Dirichlet inverse of $\,f(n) = (-1)^n$ seems to be $$ f^{-1}(n) = -\mu\!\left(n\,/\,2^{\nu_2(n)}\right)\left\lceil 2^{\nu_2(n)-1} \right\rceil, $$ where $\nu_p(n)...
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Are the extremas of $h(x)$ global?

It is well known that $li(x)$, the integral logarithm is a very good approximation of $\pi(x)$, the nunmber of primes not exceeding $x$. So, a very good approximation for the probability, that a ...
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$\forall p\in\mathbb P\exists q,r\in\mathbb P':p^3=2q+r$, $\mathbb P'=$ set of non twin primes

Define $\mathbb P'=\{n\in\mathbb P|n-2,n+2\notin \mathbb P\}$. Conjecture: Given a prime $p>3$, then $\exists q,r\in\mathbb P':p^3=2q+r.$ Tested for the first 10000 primes. The solutions ...
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Primes that are neither twin, cousin or sexy [on hold]

I'm reading up on prime pairs, and I had a question... I can't seem to find an answer to this anywhere, and the wikipedia list of prime types is enormous! Afraid I missed it when going through it. I ...
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A statement about divisibility of relatively prime integers

I'm solving a problem, and the solution makes the following statement: "The common difference of the arithmetic sequence 106, 116, 126, ..., 996 is relatively prime to 3. Therefore, given any three ...
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Why can the sieve of eratosthenes not be used to confirm the twin primes conjecture?

I have been having fun thinking about sieves and more particularly the twin prime conjecture. As I am fairly new to this type of mathematics, I am wondering, if we use the sieve of erastothenes, aka ...
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Strange results in mersenne.org database

I am interested in GIMPS project. I was browsing through known Mersenne prime numbers when I discovered strange records in their database. For example, M6972593 is the 38th Mersenne prime. However, ...
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Is there a way to translate the Sacks Spiral into a triangle?

The sacks spiral is our natural number system written in the form of a spiral and it highlights the primes which seem to fall on certain curves within the spiral. I am interested to know if there is a ...
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Generalization of Mill's theorem

Mill's theorem states that there exists a positive real number A such that the floor of the double exponential function $A^{3^n}$ are primes for all positive integers n. The value of A is ...
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A conjecture relating to Goldbach

I have a conjecture related to the strong Goldbach conjecture and the Goldbach function. It is that: for any $g(E)$, there are a finite number of even numbers which can be expressed as a sum of two ...
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What is the computational complexity of calculating $\pi(x)$ exactly?

The prime counting function $\pi(x)$ has been determined for $x=10^{26}$. The list of the $10^n$-th primes , however , ends at $n=18$. The $10^{18}$-th prime has $20$ digits. Apparantly, the ...
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Can do you repeat these calculations combining the explicit formula and Nicolas criterion, on assumption of the Riemann Hypothesis?

I did easy calculations to get for $x=N_k=\prod_{n=1}^k p_k$ the kth primorial, combining the so-called explicit formula$\dagger$ for the second Chebyshev function and Nicolas criterion for the ...
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Conjecture: Every prime number is the difference between a prime number and a power of $2$

Conjecture: $\forall p\in\mathbb P\exists q\in\mathbb P\exists n\in \mathbb N: q-p=2^n$ Verified for the 100 first primes.
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Division/Remain by a Mersenne Prime

Is it possible to compute the integer division and remainder of an integer $x$ by a Mersenne prime $p$ using only bitwise operations?
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Prove or refute that $\{p^{1/p}\}_{p\text{ prime}}$ to be equidistributed in $\mathbb{R}/\mathbb{Z}$

I've tried follow the Example 3 (see minute 30'40" of the reference), where is required the related Theorem (stated at minute 21') combined with Serre's formalism for $\mathbb{R}/\mathbb{Z}$ (also ...
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Common generator of units in finite prime fields

It is well known that the unit group of a finite field is cyclic. What can we say about the generators? Specifically I am interested in the following question: Suppose we fix a positive integer $a$, ...
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$p\in\mathbb P\iff\Big(2\leq k<\sqrt p\implies\gcd(k^2,p-k^2)=1\Big ),\;p>3$

This is sharper variant of A condition for being a prime: $\;\forall m,n\in\mathbb Z^+\!:\,p=m+n\implies \gcd(m,n)=1$ It seems enough to test that for some sums: $p=m+n\implies\gcd(m,n)=1$, namely ...
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It is possible to use the Zeta Function as primality test? [on hold]

It is possible to use the Zeta Function as primality test? $$\displaystyle\sum_{n=1}^\infty\dfrac1{n^s} = 1+\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+ ... $$ Where can I find the non-trivial zeros ...
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The sum of more than two consecutive natural numbers cannot be prime.

The sum of more than two consecutive natural numbers cannot be prime. Is the statement true and is there any way to prove it? I was able to prove that the sum of an odd amount of consecutive ...
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Asymptotic Growth of Function of Prime Counting Function

Consider $f(x)$ defined by $$f(x)=\sum_{k=1}^\infty \pi\Big{(}\frac{x}{k}\Big{)}$$ How may one another function $g(x)$ be defined such that $$\lim_{x\to\infty}\frac{f(x)}{g(x)}=1$$I have tried $g(x)=c\...
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Does the sequence $q(n)=3n+1+\frac{1-(-1)^n}{2}$ generate all the prime numbers?

Define $$q(n)=3n+1+\frac{1-(-1)^n}{2} \quad, \quad n\in \mathbb N.$$ $$1,5,7,11,13,17,19,23,25,29,31,35,\dots$$ It seems like this formula gives all primes $>3$ (although not just primes of ...
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For which $0\leq a<p^2$, where $p$ is an odd prime, we have that $(2p-1)!\equiv a\mod{p^2}$

Let $p$ be an odd prime. I need to find for which $0\leq a < p^2$, $(2p-1)!\equiv a\mod{p^2}$. If $a\equiv (2p-1)!\mod{p^2}$, then we have that $a = kp^2 + (2p-1)!$, and therefore $p\mid a$, ...
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Prime Number Theorem and the Riemann Zeta Function

Let $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$$ be the Riemann zeta function. The fact that we can analytically extend this to all of $\mathbb{C}$ and can find a zero free region to the left of the ...
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Show that for $x,y,z\in\mathbb{Z}$, if $x$ and $y$ are coprime, then $\exists n\in\mathbb{Z}$ such that $z$ and $y+xn$ are coprime.

Not sure where to start on this one. I understand that coprime indicates that their GCD is 1, but I am somewhat confused how to proceed.
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Is one of $k+1^2,$ $k+2^2,$ …, $k+N^2$ always prime?

I know that the Bunyakovsky conjecture is still open, so we can't prove that there exist primes of the form $n^2+k$ for a given $k$. But suppose that they do: is the least $n$ such that $n^2+k$ is ...
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An unexplained condition on $a$ in a proof on the primes?

Lemma A positive integer $n$ is a prime if $(n,p) = 1$ for every prime integer $p \leq \sqrt{n}$ Proof in my text Let $(n,p) = 1$ for every prime $p \leq \sqrt{n} \:$. Suppose $n$ is not a ...
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Is the error I noticed a harmless typo?

Here http://arxiv.org/PS_cache/arxiv/pdf/1002/1002.0442v1.pdf , at page $2$ at the bottom, it is stated that the number of primes not exceeding $x$, denoted by $\pi(x)$, satisfies the double-...
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Prove that $\sum^{n-1}_{i=1}i^{(n-1)} \equiv -1$ (mod $n$) for all prime $n\in\mathbb{N}$.

Prove that $\sum^{n-1}_{i=1}i^{(n-1)} \equiv -1$ (mod $n$) for all prime $n\in\mathbb{N}$. I'm having a difficult time proving this problem. I was able to verify that it works for prime $n$ up to ...
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Does every power of two arise as the difference of two primes?

Conjecture: For each $n\in\mathbb N$ there are primes $q<p$ with $p-q=2^n$. Verified for $n\leq 26$: ...
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If $q\mid 2^p + 3^p$ then $q \gt p$

Let $p, q$ positive prime numbers, $q > 5$. Prove that if $q \mid \left(2^{p} + 3^{p}\right)$ then $q > p$. First, it's clear that $p \ne q$ because, using Fermat's little theorem, $2^p = ...
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Are all theorems usable? [closed]

The (revised) question to answer: Can anyone give an example of a serious proof using this funny (revised) theorem? For any natural number $n$ and prime $p<n-1$ there exist a prime $q$ ...
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Growth of $\pi(2x) - 2\pi(x)$

In Hardy & Wright's Theory of Numbers (p. 494f in 6th ed.) there's a little discussion following the proof of the prime number theorem. We have $$ \pi(2x) - \pi(x) = \frac{x}{\log x} + o\...
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Is this a known conjecture? Given odd primes $p,q$ with $p + q$ sufficiently large, must there exist a different pair $p',q'$ with $p+q = p'+q'$?

Conjecture: There is a natural number $N\in\mathbb N$ such that given odd primes $p,q$ with $p+q>N$ there are primes $p',q'$ where $p' \notin \{p,q\}$ such that $p+q=p'+q'$. Is this known?
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Statement regarding primes $ \le n$

Following is the statement I believe is true, but can't prove. Let $n$ be a natural. Let the primes less than equal to $\sqrt{n}$ be $p_1,p_2,...,p_k$. Let $\alpha_i$ be the greatest natural ...
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Simpler way to compress this exponentiation?

I am trying to find when the following is true: Let $H =(10k)^b \bmod 6(p-1)$ Let $J = 10^{H} \bmod 9p$ For some prime $p > 5$ and large $k,b$. I am trying to find when $J$ is equal to $1$. ...
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A function can provide the complete set of Euler primes via a Mill's-like constant. Is it useful or just a curiosity?

The following $f(m,n)$ function provides the complete set of Euler primes (OEIS A196230): $$f(m,n)=m^2-m+[\lfloor E^{2^n} \rfloor - {\lfloor E^{2^{n-1}} \rfloor}^2 +\frac{\lvert n-(\frac{1}{2}) \...
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Help on an application of Dirichlet's theorem for primes in progression

Suppose that I have an infinite sequence of positive integers $$a_1,\ldots,a_m,\ldots$$ with the following recursion $$a_{m+1} -a_m =b(m+1)$$ So that $$a_{m+1} =b(m+1) +a_m$$ Suppose ...
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Number of prime exponents for Generalized Mersenne Primes

Please help me on the following scenario(s): Estimate the number of primes $p$ less than or equal to $x$ such that there is a prime of the form ${(a+1)}^p$ $-$ $a^p$ for all $a$ < $50$? What is ...
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is the multiplication of n consecutive prime numbers starting with 2 plus 1 prime?

The question kinda tell everything for itself, let P(n) be the n-th prime number, is $(\Pi_1^n P_n)+1$ prime ?
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On Dirichlet series and Firoozbakht's conjecture

On assumption of the Firoozbakht's conjecture (this is the Wikipedia, but the reference is for Carlos Rivera's Page) one has that can writes informally the Dirichlet series in LHS of this inequality $$...
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Primes in the binomial transform of $ [1, 1, 2, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, …]$.

This question is related to this sequence A139482. A commentator gives the following formula for $a_m$ $$a_m = {3m^2-9m+10 \above 1.5pt 2}$$ I have that you should consider the sequence $b_n =3n+2$ ...
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Is a tight concrete bound for the error-term in the prime-number-theorem known?

Here : https://en.wikipedia.org/wiki/Prime_number_theorem it is mentioned that $$\pi(x)=Li(x)+O(xe^{-a\sqrt{ln(x)}})$$ What is a tight upper bound for $|\pi(x)-Li(x)|$ in concrete terms ? The ...
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Two randomly chosen coprime integers

This is a twist on the problem commonly known to have solution $6/\pi^2$. Suppose when choosing from all natural numbers $\mathbb{N}$, the probability of choosing $n \in \mathbb{N}$ is given by $P(n)=...
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characterisation of $n$ as prime using min values of $x$ such that $nx+1$ or $nx$ is square

Let $n\ge 5$ be an odd integer and $k\ =\ \min\{x\in\mathbb{N}\colon nx+1\text{ is a perfect square}\}$ $l\ =\ \min\{x\in\mathbb{N}\colon nx\text{ is a perfect square}\}$ Prove that $n$ is a prime ...
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a consequence of Prime Number Theorem

By Prime Number Theorem we have $\lim_{n\to\infty}\frac{p_{n+1}}{p_n}=1$, so $\frac{p_{n+1}}{p_n}=1+a_n$ where $a_n\to 0$. How fast does $(a_n)$ converge to $0$ ? Does for example $a_n\ln n$ or $...
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Generalization of Inkeri's primality test

How to prove that following hypothesis is true ? 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 ...
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Longest sequence of primes where each term is obtained by appending a new digit to the previous term

What is the longest known sequence of primes where each new term is obtained by appending a new decimal digit to the previous term? Examples: $$(2,23,233,2333,23333)$$ There are no more members in ...
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How to test if $n!+1$ is prime or not?

for $n=0,1,2,3,11,27,37,41,73,77,116,154,320,340,399,427,872,1477,6380,26951,...$ $$n!+1$$ is prime. But how can you proof (with 100% certantiy) thats the case? Especially for the larger ones. For ...
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For every prime $p > 3$ that is $3$ mod $4$, does $q+1 \mid p-q$ for some other prime $q$?

Yet another random conjecture about primes: Given a prime $p>3$ of the form $4n+3$. Then there exist a prime $q<p$ such that $q+1\mid p-q$. Verified for all $p<100000$.