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Prime numbers

Table of contents: Basic formulas - Numerical values - Bounds and inequalities

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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\!\left(x\right) Prime counting function
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Symbol: RiemannHypothesis RH\operatorname{RH} Riemann hypothesis

Basic 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:pPandpx}\pi\!\left(x\right) = \# \left\{ p : p \in \mathbb{P} \,\mathbin{\operatorname{and}}\, p \le x \right\}

Numerical values

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Table of pnp_{n} for 1n2001 \le n \le 200

Bounds and inequalities

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pn+1<2pnp_{n + 1} \lt 2 p_{n}
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π ⁣(2x)π ⁣(x)1\pi\!\left(2 x\right) - \pi\!\left(x\right) \ge 1
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pn>n(log ⁣(nlog ⁣(n))1)p_{n} \gt n \left(\log\!\left(n \log\!\left(n\right)\right) - 1\right)
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pn<nlog ⁣(nlog ⁣(n))p_{n} \lt n \log\!\left(n \log\!\left(n\right)\right)
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π ⁣(x)>xlog ⁣(x)\pi\!\left(x\right) \gt \frac{x}{\log\!\left(x\right)}
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π ⁣(x)<1.25506xlog ⁣(x)\pi\!\left(x\right) \lt \frac{1.25506 x}{\log\!\left(x\right)}
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π ⁣(x)li ⁣(x)<xlog ⁣(x)8π\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.

2019-07-15 23:42:41.550119 UTC