Prime numbers are natural numbers 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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On Goldbach conjecture

Let $N$ a large natural number, let $\forall n\leq N,\, R_{2}\left(n\right)=\underset{p_{1}+p_{2}=n}{\sum}\log\left(p_{1}\right)\log\left(p_{2}\right)$ and let $S\left(\alpha\right)=\underset{p\leq ...
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Is there any relation between prime numbers and $\sqrt 2$?

Is there any relation between prime numbers and $\sqrt 2$? Connection between primes and $e$ , $\pi$ are obvious and can be googled easily but unable to find connection between any square root.
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Riemann prime counting function / Log Integral

I include the beginnings of an investigation: $$\text{A plot of R}(x)\text{ against }\pi(x):$$ $$\text{A plot of li}(x)\text{ against }\sum_{n=1}^{x}\frac{\pi(x^{1/n})}{n}:$$ It seems as though ...
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If there is an integer $n$ such that $n^2\equiv3\pmod p$, where $p$ is prime, prove there are integers $a$ and $b$ such that $|a^2-3b^2|=p$

If there is an integer $n$ such that $n^2\equiv3\pmod p$, where $p$ is prime, prove there are integers $a$ and $b$ such that $|a^2-3b^2|=p$. So $n^2-3 = pm$ for some integer $m$, and I know $|a^2 ...
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$\frac{ra}{p} + \frac{rb}{p} + \frac{rc}{p} + \frac{rd}{p} = 2 $, with $p$ prime

Let $p>2$ be a prime and let $a$, $b$, $c$, $d$ be integers not divisible by $p$, such that $\{\frac{ra}{p}\} + \{\frac{rb}{p}\} + \{\frac{rc}{p}\} + \{\frac{rd}{p}\} = 2 $ for any integer $r$ not ...
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Liouville function and PNT

The Big Omega function is defined as the number on non-distinct prime factors of an integer. I.e. $\Omega (2^a3^b...p^z)=a+b+...+z$, and the Liouville function is defined as ...
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Special matrices with determinant 0

Define a quadratic matrix A with n rows and n columns by filling it with consecutive primes, starting with some prime p. The object is, to find the least starting prime p, such that A has determinant ...
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Conjecture similar to Dirichlet's theorem

Dirchlet's theorem states that there are infinitely many primes of the form an+b , where n is a natural number, when gcd(a,b)=1. Let a number which is the product of two distinct primes with the ...
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Instantly Factor a Semiprime of Any Size?

recently I've been looking at the interesting problem of RSA encryption and attempting to understand what it's so hard to find the factors then I came across this page. Can anyone comment on it's ...
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Global maximum for $\log x - \frac{x}{\pi(x)}$

$\log x - \frac{x}{\pi(x)}$ hits a global maximum at $x=24,137$ with a value of $1.11196\dots$ Is there any documentation about this anywhere? I couldn't find any. Apologies if there is and it is ...
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When is a number like “ddd…ddd”+1 (where d is a digit) a perfect square or a prime?

Inspired by Is the number $333, 333, 333, 333, 333, 333, 333, 333, 334$ a perfect square?, I wonder when numbers like these are perfect squares. Certainly, all numbers of the form $000...0001$ are ...
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The intersection of two sets of natural numbers defined via primes is infinite

Any given prime $P \in \mathbb P$ is either of the form $$ P \equiv -1 \mod 6$$ or $$ P \equiv 1 \mod 6.$$ In other words, every prime greater than 3 can be expressed in the form $$P = 5+6n \vee 7 + ...
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Full reptend prime generation

I'm pretty undereducated when it comes to mathematics but I'm working on that! At the moment, I am trying to generate full reptend prime numbers and I'm having some problems (I know they can be found ...
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Explicit form for $\sum_p \ln(\ln(p))$?

Riemann gave an explicit form for the counting function of the primes. Is there an explicit form for the counting function $f(x) = \sum_p \ln(\ln(p))$ where the sum is over $p$ : the number of primes ...
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An upper bound on the least common multiple of the first $2n+1$ integers

Let $p$ be a prime number and let $a, n \in \mathbb{N}$. Then $$ p^a \mid \operatorname{lcm}(1, 2, \dots, 2n+1) \implies p^a \leq 2n + 1 \implies a \leq \dfrac{\ln(2n+1)}{\ln p}$$ and ...
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Wilson's theorem intuition

Wilson's Theorem: $p$ is prime $\iff$ $(p-1)!\equiv -1\mod p$ I can use Wilson's theorem in questions, and I can follow the proof whereby factors of $(p-1)!$ are paired up with their (mod $p$) ...
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Number of primes of type 4*n +1 in a range

I want to find number of primes which are congruent 1 (mod 4) in a range [a, b]. The range can be of order $10^9$ as a and b can be from $1$ to $10^9$. I tried segmented sieve but for a range so ...
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Approximation to $\pi(x)$ conjecture.

A friend conjectured that $\left[\prod_{k=1}^{a_j <\sqrt{x}} \left(1-\frac{1}{a_k}\right)\right] x$ is usually closer to $\pi(x)$ than $\operatorname{Li}(x)$ is for some (fixed) sequence of ...
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Related almost primes

Do there exist positive integers $w, x, y, z$ and an infinite set $A$ such that for all $a \in A$, $a$ is a $w$-almost prime and $b = y \cdot a + z$ is an $x$-almost prime? With $(w,x,y,z) = ...
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How to investigate whether number is prime

Investigate whether 36611 is prime. Show the first few steps of the procedure. This is the question. Is it enough to show its not divisible by 2, 3, 5 ? Surely this must not be enough. But how do i ...
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Prime number finding via polynomials

I try to find approximation polynomial to estimate which number is prime or not. Addtion to this, (If It is possible) To find the closed form of coefficients of the series ($c(n)$) Euler found the ...
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Does the sum over the normed differential of the prime power function equal $2\log2\pi$?

Let $p\in \Bbb P$ a prime and a prime power function: $$\xi_p(x) = p^x$$ with $x \in \Bbb R^+_0$ hence: $$\xi'_p = \frac{d}{dx}\xi_p=\xi_p \log p$$ Taking into account E. Muñoz García and R. ...
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Are there infinitely many primes $p_k$ such that $(p_k-1)!+p_k$ is a also prime?

I am wondering whether there are infinitely many primes $p_k$ such that $(p_k-1)!+p_k$ is also prime. Given that $p_k \equiv 2 \pmod 3$. For a very large prime, I can assume Stirling's ...
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Merthen's third theorem and uncertainty of prime hits

Conjecture(1) Merten's third theorem says: $$\lim_{L\to\infty}\ln L\prod_{p\le L}\left(1-\frac1p\right)=e^{-\gamma}$$ we have a wild discussion here around the table whether it is possible to ...
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Could a determinstic primality test specialized to this form of prime exist?

Is it possible there could be an "efficient" deterministic primality test for prime numbers of the form $$(2^n + 1)^2 - 2$$ or $$(2^n - 1)^2 - 2$$ in the same vein as the Lucas-Lehmer test for ...
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Unique decomposition of $c$ sums of products of $k$ numbers greater than 1, allowing duplicates?

This question differs from Unique decomposition of $c$ sums of products of $k$ prime numbers, allowing duplicates? in that prime number restriction is changed to any number greater than 1. Suppose ...
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Prime numbers with binomial coefficients

Let $p$ be an odd prime and $n$ a positive integer. Prove that $p+1$ divides $n$ if and only if $$\sum_{k\equiv j\pmod{p-1}}^n\binom{n}{k}(-1)^{\frac{(k-j)}{p-1}}\equiv 0 \mod p$$ for every $$j\in ...
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Definition of a complex prime

If there would be such a thing as a complex prime number, how would it be defined? A normal prime is defined as a number only dividable by one or itself.. would that be "1+1i" with complex numbers? ...
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Prove $\left(\frac{q(q+1)}{p}\right) =\left(\frac{1+q^{-1}}{p}\right )$ for $p\gt2$ a prime, and any $q \in \mathbb{Z^+} $.

For $p\gt2$ a prime, and any $q \in \mathbb{Z^+} $, Show that $\left(\frac{q(q+1)}{p}\right) =\left(\frac{1+q^{-1}}{p}\right )$ where the terms are legendre terms. I saw this result as part of a ...
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For which positive integers n does there exist a prime whose digits sum to n?

Motivated by this earlier question, I thought of this problem: Question: For which positive integers $n$ does there exist a prime whose decimal digits sum to $n$? We can make two "easy" ...
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$\sum_p z^p$ where $p$ is prime

I've started reading Shakarchi's Complex Analysis, and I thought about something interesting. If I haven't mistaken, for any subsequence $A\subset \mathbb{Z}^+$, $\sum_{n\in A} z^n$ has radius of ...
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Searching for prime candidates

For some additional excitement, I've been searching for primes $p \gg q = 104729$, where $q$ is of course the ten-thousandth prime. It seems that the best way to search for prime candidates $p$ is to ...
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How many co-primes are there for a big integer N over a specified interval?

How many co-primes are there for a big integer $N$ over a specified interval ? bounds of $N$ are $[1,10^9]$ and the interval is $[a,b]$ where ($1\leq a\leq b \leq 10 ^{15}$) and there are $100$ ...
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Solving $key=(\sum_{K=0}^n\frac{1}{a^K})\mod m$ with High limits

I was solving this equation:- $$key=(\sum_{K=0}^n\frac{1}{a^K})\mod m$$ Given $$ 1,000,000,000 < a, n, m \; < 5,000,000,000 $$ $$ a, m \; are \;coprime $$ I solved it bruteforcely but it ...
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asymtotic ratio of nonsquarefree repunits

Let $R_n:=\frac{10^n-1}{10-1}$ (called a repunit) and $\mu$ be the Moebius function. Also $[n]:=\{1,2,3,\cdots, n\}, A_n:=\{m \in [n]| \mu (R_m)=0\}.$ What is the value of $\lim \limits_{n ...
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Integral with Sums of Prime Counting Functions

I came across the following integral, while working with products of $\zeta$ primes function: $$ \int_{1}^{x}t^{-s-1} \sum_{i=1}^{\pi(t^{1/2})}\left[\pi\left(\frac{t}{p_i}\right)-i+1\right] dt, $$ ...
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What is a good tool for this job involving the prime spiral?

I'm interested in studying the prime spiral interactively. This question talks about some interesting patterns in the spiral involving quadratic equations. The idea I had was, write a program that ...
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Prime numbers of the form: $k\cdot 2^n \pm 1$ , where $k<3n$

Is it true that : For every $n$ there exists a number $k<3n$ such that: $k\cdot 2^n-1$ or $k\cdot 2^n+1$ is prime,where $k,n\in \mathbf{N}$ Maple code that prints least $k$ such that ...
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Prime numbers of the form : $2^{n+a}+2^{n} \pm 1$ , where $0 \leq a < n$ and $n \equiv 0 \pmod 6$

Is it true that : For any positive integer $n$ such that $n \equiv 0 \pmod 6$ there is at least one prime number of the form: $p=2^{n+a}+2^{n} + 1$ , or , $p=2^{n+a}+2^{n} - 1$ with ...
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$n$ by $n$ Primally Magic Squares

(Again copied verbatim from a September 2009 thread I made.) A Primally Magic Square (PMS) is exactly like a traditional magic square with a change of criteria. Where a traditional magic square is ...
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Calculating $n$-th $q:P(q)=p \in \Bbb P$

Let $P(x)$ denote the number of ways of writing an integer $x$ as a sum of positive integers (where permutation of the array of integers in the sum doesn't count). Ex: $P(1)=1, P(2)=2,P(4)=5$. Let ...
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Are logarithms of prime numbers algebraically independent?

From Baker's theorem it follows that a linear combination of natural logarithms of prime numbers with non-zero algebraic coefficients can never be zero. Has it been proved that the set of all natural ...
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The asymptotic behaviour of triples $n!+q^{n!}=c$, where $q$ is the first prime greater than $n$, and abc conjecture

For a large positive integer $n$, let $q=q(n)$ (below we denote this $q=q(n)$ by $q_{N}$ because we assumed that is the $N$-th prime number) the first prime number which is (strictly) greater that ...
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Probability number is divisible by half the square of a prime?

Let $p$ be a prime. What is the probability that a number of the form $\left \lceil \frac{p^2}{2} \right \rceil$ divides a random positive integer $n$. I have a solution that involves the Riemann-Zeta ...
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sextic reciprocity and divisibility question

Regarding the question if $p|(2^{2(p-1)/6}+2^{(p-1)/6}+1) $ where $p$ is a prime of the form $7\mod 8 $ That is how far I got: $2^{(p-1)/6} \mod\ p\equiv x $ if the solution of $x^6\ mod\ ...
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Given a number $N$ and a prime $P$, how many numbers $\leq N$ are divisable by P but not by any smaller primes?

The following Math Exchange question deals with a similar problem: not divisible by 2,3 or 5 but divisible by 7 However, the answers given become infeasible quite quickly because the amount of ...
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$p$ be a prime number , then is it true that $p^2| {pa \choose pb}-{a \choose b} , \forall a,b \in \mathbb N$ ?

Let $p$ be a prime number , then is it true that $p^2| {pa \choose pb}-{a \choose b} , \forall a,b \in \mathbb N$ ?
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Is it true that $(1+\varepsilon)\pi(x+y)\ge\pi(x)+\pi(y)$?

Recently I was going through Udrescu's result concerning the inequality $\pi(x)+\pi(y)\ge\pi(x+y)$ which states that ($\pi(x)$ denotes the prime counting function), For all $\varepsilon>0$ and ...
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Which fields of math would I need to study to fully understand/solve the Riemann Hypothesis?

Apart from analytic number theory / complex analysis to actually knowing what it's about, which fields of math should I master to have a chance at solving it? I understand that the answer may come ...
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Number of binomial coefficients , ${ n \choose k}$ k $\in$ [0,n] , that are divisible by a prime p?

For a given k, ${n\choose k}$ is divisible by a prime p if and only if at least one of the base p digits of n is greater than the corresponding base p digit of k (consider the p-ary notation for n ...