The first Skewes Number, written \(Sk_1\), is the smallest number \(n\) such that \(\pi(n) > li(n)\) is true, assuming the Riemann hypothesis is true. Here, \(\pi(n)\) is the prime counting function and \(li(n)\) is the logarithmic integral.[1]
The original upper bound for the number was \(e^{e^{e^{79}}} \approx 10^{10^{10^{34}}}\).
The second Skewes Number, \(Sk_2\), is the smallest number \(n\) such that \(\pi(n) > li(n)\) is true, assuming the Riemann hypothesis is false. It is larger than the original Skewes Number, \(e^{e^{e^{e^{7.705}}}}\) ~ \(10^{10^{10^{963}}}\).
Leading digits of exponent
We can't actually compute leading digits of both Skewes' numbers, but can calculate first digits of "number of digits". The following transformation shows this (for \(Sk_1\)):
\(e^{e^{e^{79}}} = e^{10^{e^{79} \times log(e)}} = 10^{10^{e^{79} \times log(e)} \times log(e)} = 10^{10^{e^{79} \times log(e)+log(log(e))}}\). On the big number calculator: \(10^{e^{79} \times log(e)+log(log(e))} = 35536897484442193330...\), so \(Sk_1 = 10^{35536897484442193330...}\)
Also, similarly, \(Sk_2 = 10^{29377275332206251151...}\)