# Prime numbers

Symbol: PP $\mathbb{P}$ Prime numbers
Symbol: PrimeNumber $p_{n}$ nth prime number
Symbol: PrimePi $\pi\!\left(x\right)$ Prime counting function
Symbol: RiemannHypothesis $\operatorname{RiemannHypothesis}$ Riemann hypothesis

## Basic formulas

$\mathbb{P} = \left\{ p_{n} : n \in \mathbb{Z}_{\ge 1} \right\}$
$\pi\!\left(x\right) = \left|\left\{ p : p \in \mathbb{P} \,\mathbin{\operatorname{and}}\, p \le x \right\}\right|$

## Numerical values

Table of $p_{n}$ for $1 \le n \le 200$

## Bounds and inequalities

$p_{n + 1} \lt 2 p_{n}$
$\pi\!\left(2 x\right) - \pi\!\left(x\right) \ge 1$
$p_{n} \gt n \left(\log\!\left(n \log\!\left(n\right)\right) - 1\right)$
$p_{n} \lt n \log\!\left(n \log\!\left(n\right)\right)$
$\pi\!\left(x\right) \gt \frac{x}{\log\!\left(x\right)}$
$\pi\!\left(x\right) \lt \frac{1.25506 x}{\log\!\left(x\right)}$
$\left|\pi\!\left(x\right) - \operatorname{li}\!\left(x\right)\right| \lt \frac{\sqrt{x} \log\!\left(x\right)}{8 \pi}$

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-06-18 07:49:59.356594 UTC