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Fungrim entry: bfa464

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)
Assumptions:nZ46254381n \in \mathbb{Z}_{\ge 46254381}
References:
  • https://arxiv.org/abs/1706.03651
TeX:
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)

n \in \mathbb{Z}_{\ge 46254381}
Definitions:
Fungrim symbol Notation Short description
PrimeNumberpnp_{n} nth prime number
Loglog(z)\log(z) Natural logarithm
Powab{a}^{b} Power
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("bfa464"),
    Formula(Less(PrimeNumber(n), Mul(n, Sub(Add(Sub(Add(Log(n), Log(Log(n))), 1), Div(Sub(Log(Log(n)), 2), Log(n))), Div(Add(Sub(Pow(Log(Log(n)), 2), Mul(6, Log(Log(n)))), Decimal("10.667")), Mul(2, Pow(Log(n), 2))))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(46254381))),
    References("https://arxiv.org/abs/1706.03651"))

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2021-03-15 19:12:00.328586 UTC