# Tagged Questions

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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### Is any elementary proof important (beside Selberg's work) ?

Is any elementary proof important (beside Selberg's work) ? Plus, why is the elementary proof of prime number theory by Selberg so important ? Selberg was awarded the Field medal is mainly because ...
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### Conditions for which $(\sum p_i)(\sum \frac{1}{p_i})$ over an arbitrary $i$ for a set of primes $\{p_i\}$ is unique?

I am looking for conditions (if any are needed beyond properties of primes) for which $(\sum p_i)(\sum \frac{1}{p_i})$ over an arbitrary $i$ for a set of primes $\{p_i\}$ is unique in that there is ...
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### A slightly various form of Dirichlet's theorem on arithmetic progressions

Are there infinitely many primes of the form $2n(n+1)+1$?
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### Are there infinitely many prime numbers in $a_n=\frac{7\times 10^n-1}{3}$?

In the array $a_n=\frac{7\times 10^n-1}{3}$, are there infinitely many primes? (when $n={7+16k},a_n$ is divisible by $17$, so there are infinitely numbers not prime)
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### The fastest way to count prime number that smaller or equal N

I want to count all prime numbers that existing in N but I don't know how to count. Can any one tell me how to count prime numbers that are smaller than or equal to N in mathematics formal?
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### Possible Prime Sum Pattern (Amateur)

Disclaimer: I’m an amateur, and have no advanced knowledge of math, so please forgive my ignorance as I’m just curious to know if I’ve stumbled upon something or not. Prime Numbers: 2, 3, 5, 7, 11, ...
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### Does $\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ converge to the golden ratio?

The sum $\displaystyle\sum\limits_{n=2}^{\infty}\frac{1}{n\ln(n)}$ does not converge. But the sum $\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ where $P_n$ denotes the $n$th prime ...
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### Sequence of primes by concatenating digits in a given base.

Given a base, $b$ is there is a sequence $\lbrace a_n\rbrace_{n\geq 0}$ where $a_k \in \lbrace 1,2\cdots, b-1\rbrace$so that the sequence: $$b_n:= \sum_{k=0}^n a_kb^k$$ is a sequence of primes ...
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### Digit wise modulo for calculating power function for very very large positive integers

I am writing a code to calculate $P^Q$ where $P$, $Q$ are positive integers which can have number of digits up to $100000$. I want the result as $r = P^Q \pmod{10^9+7}$, where $10^9+7$ is a prime ...
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### How do you go about finding a 12 digit prime number?

How do you go about finding a 12 digit prime number?
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### Finitely many Supreme Primes?

A challenge on codegolf.stackexchange is to find the highest "supreme" prime: http://codegolf.stackexchange.com/questions/35441/find-the-largest-prime-whose-length-sum-and-product-is-prime A supreme ...
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### Divisibility of $a^p-r$ and $a^q-r$ by the primes $p,q$
Let $p, q$ be prime and $a$ some positive integer such that $a = pq + r$ where $r$ is the remainder. Show that $p \mid a^p – r$ and $q \mid a^q – r$. Example: $p = 3$ and $q = 5$, $a = 17$ and \$r = ...