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Prime numbers

Table of contents: Definitions - Connection formulas - Tables - Bounds and inequalities

Definitions

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Symbol: PP P\mathbb{P} Prime numbers
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Symbol: PrimeNumber pnp_{n} nth prime number
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Symbol: PrimePi π(x)\pi(x) Prime counting function
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Symbol: RiemannHypothesis RH\operatorname{RH} Riemann hypothesis

Connection formulas

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P={pn:nZ1}\mathbb{P} = \left\{ p_{n} : n \in \mathbb{Z}_{\ge 1} \right\}
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π(x)=#{p:pP  and  px}\pi(x) = \# \left\{ p : p \in \mathbb{P} \;\mathbin{\operatorname{and}}\; p \le x \right\}
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pn=A000040 ⁣(n)p_{n} = \text{A000040}\!\left(n\right)
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π(n)=A000720 ⁣(n)\pi(n) = \text{A000720}\!\left(n\right)

Tables

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Table of pnp_{n} for 1n2001 \le n \le 200
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Table of p10np_{{10}^{n}} for 0n240 \le n \le 24
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Table of π ⁣(10n)\pi\!\left({10}^{n}\right) for 0n270 \le n \le 27

Bounds and inequalities

Bertrand's postulate

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pn+1<2pnp_{n + 1} < 2 p_{n}
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π ⁣(2x)π(x)1\pi\!\left(2 x\right) - \pi(x) \ge 1

Bounds for prime numbers

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pn<nlog ⁣(nlog(n))p_{n} < n \log\!\left(n \log(n)\right)
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pn>n(log ⁣(nlog(n))1)p_{n} > n \left(\log\!\left(n \log(n)\right) - 1\right)
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pn<n(log(n)+log ⁣(log(n))1+log ⁣(log(n))2log(n)log2 ⁣(log(n))6log ⁣(log(n))+10.6672log2 ⁣(n))p_{n} < n \left(\log(n) + \log\!\left(\log(n)\right) - 1 + \frac{\log\!\left(\log(n)\right) - 2}{\log(n)} - \frac{\log^{2}\!\left(\log(n)\right) - 6 \log\!\left(\log(n)\right) + 10.667}{2 \log^{2}\!\left(n\right)}\right)
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pn>n(log(n)+log ⁣(log(n))1+log ⁣(log(n))2log(n)log2 ⁣(log(n))6log ⁣(log(n))+11.5082log2 ⁣(n))p_{n} > n \left(\log(n) + \log\!\left(\log(n)\right) - 1 + \frac{\log\!\left(\log(n)\right) - 2}{\log(n)} - \frac{\log^{2}\!\left(\log(n)\right) - 6 \log\!\left(\log(n)\right) + 11.508}{2 \log^{2}\!\left(n\right)}\right)

Bounds for the prime counting function

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π(x)<1.25506xlog(x)\pi(x) < \frac{1.25506 x}{\log(x)}
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π(x)li(x)<xlog(x)8π\left|\pi(x) - \operatorname{li}(x)\right| < \frac{\sqrt{x} \log(x)}{8 \pi}
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π(x)>xlog(x)\pi(x) > \frac{x}{\log(x)}

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

2020-04-08 16:14:44.404316 UTC