About: Skewes number   Sponge Permalink

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The first Skewes number, written \(Sk_1\), is an upper bound for the least number \(n\) such that \(\pi(n) > li(n)\) is true, where \(\pi(n)\) is the prime counting function and \(li(n)\) is the logarithmic integral. This bound was first proven assuming the Riemann hypothesis. It's equal to \(e^{e^{e^{79}}} \approx 10^{10^{10^{34}}}\). As of now, it is known that the least example \(n\) of \(\pi(n) > li(n)\) must lie between \(10^{14}\) and \(1.4 \cdot 10^{316}\).

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  • Skewes number
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  • The first Skewes number, written \(Sk_1\), is an upper bound for the least number \(n\) such that \(\pi(n) > li(n)\) is true, where \(\pi(n)\) is the prime counting function and \(li(n)\) is the logarithmic integral. This bound was first proven assuming the Riemann hypothesis. It's equal to \(e^{e^{e^{79}}} \approx 10^{10^{10^{34}}}\). As of now, it is known that the least example \(n\) of \(\pi(n) > li(n)\) must lie between \(10^{14}\) and \(1.4 \cdot 10^{316}\).
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abstract
  • The first Skewes number, written \(Sk_1\), is an upper bound for the least number \(n\) such that \(\pi(n) > li(n)\) is true, where \(\pi(n)\) is the prime counting function and \(li(n)\) is the logarithmic integral. This bound was first proven assuming the Riemann hypothesis. It's equal to \(e^{e^{e^{79}}} \approx 10^{10^{10^{34}}}\). The second Skewes number, \(Sk_2\), is a closely related upper bound for the least number \(n\) such that \(\pi(n) > li(n)\) holds, but this bound, as opposed to the previous one, was proven without assumption of the Riemann hypothesis. It is equal to \(e^{e^{e^{e^{7.705}}}}\) ~ \(10^{10^{10^{963}}}\), which is larger than the original Skewes number. As of now, it is known that the least example \(n\) of \(\pi(n) > li(n)\) must lie between \(10^{14}\) and \(1.4 \cdot 10^{316}\).
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