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Apocalyptic number

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Not to be confused with Apocalypse number.

Apocalyptic numbers are numbers of the form 2n containing the digits 666 in their decimal expansions.[1] 2157 is the smallest apocalyptic number:

182687704666362864775460604089535377456991567872

2n is an apocalyptic number for n = 157, 192, 218, 220, 222, ... (OEIS A007356). These values of n become increasingly dense, and as \(n \rightarrow \infty\), the probability of \(2^n\) being apocalyptic becomes 1. Thus, when sufficiently large, apocalyptic numbers cease to be interesting and non-apocalyptic powers of two become more of a novelty.

There are 3716 non-apocalyptic numbers of the form 2n for \(0 \le n \le 1000000\), the largest of which is \(2^{29784}\). From heuristic considerations, \(2^{29784}\) is very likely the largest one.

Specific numbers Edit

The apocalyptic number 2220 is particularly interesting, being the smallest one containing 666 twice:

1684996666696914987166688442938726917102321526408785780068975640576

It also has the first set of five consecutive sixes.

\(2^{11666}\) and \(2^{26667}\) are two non-apocalyptic numbers that contain 666 in their base-2 logarithms. There is an overwhelming probability that there are not any others.

Tetrational apocalyptic numbers Edit

The first apocalyptic number in the form \(^n2\) is \(^52\). Since the last digits of \(^n2\) converge, a 666 will almost surely freeze at some point into the convergent digits. In fact one does so at \(n = 1213\), and there is a finite (but unknown) number of non-apocalyptic numbers of this form. If \(2^{29784}\) really is the largest non-apocalyptic power of 2, then \(^42\) is the largest non-apocalyptic power tower of 2.

Sources Edit

  1. Apocalyptic Number -- from Wolfram MathWorld

See alsoEdit

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