# Questions tagged [prime-numbers]

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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### Primes $p$ for which $2p \pm 1$ are also primes

Out of curiosity and trouble sleeping, I decided to look at the distribution of primes $p$ for which $2p \pm 1$ are also primes. I looked at the first 25,910,000 primes and counted the number of ...
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### Relatives of Heegner numbers?

It is well known that Euler's lucky numbers are related to the Heegner numbers, where \begin{align} &n^2+n+p\\ \end{align} gives primes for $n=0,\dots,p-2$ if and only if its discriminant $1-4p$...
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### The effect of roots of Dirichlet's $\beta$ function condenses to $\frac12\left(1+ie^{i2\pi\frac{p}4}\right)$

With the help of Raymond Manzoni and Greg Martin I was able to derive an explicit formula for the number of primes of the form $4n+3$ in terms of (sums of) sums of Riemann's $R$ functions over roots ...
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### Prime numbers, analysis of polylogarithms

Can any interesting results concering prime numbers be obtained using the analytic properties of the polylogarithm, similar to how analytic methods are used on the zeta function to obtain results ...
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### Pairs of palindromic primes without $1$ and have a palindromic product

While discussing about prime numbers with other users, I noticed that: $(1)$ There are very few pairs of palindromic prime numbers that do not contain the digit $1$ and that have products which are ...
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### The Existence of “Simple” Prime Generating Functions

Obviously, we do not know an explicit and easily manipulable formula for finding every prime - that is, a function $f(n)$ which yields the $n^{th}$ prime. I've seen plenty of formulas that "cheat" in ...