Apocalyptic number

Apocalyptic numbers are numbers of the form containing the digits 666 in their decimal expansions.Apocalyptic Number -- from Wolfram MathWorld is the smallest apocalyptic number:
 is an apocalyptic number for n = 157, 192, 218, 220, 222, ... (). 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 3,716 non-apocalyptic numbers of the form  for \(0 \le n \le 1,000,000\), the largest of which is \(2^{29,784}\). From heuristic considerations, \(2^{29,784}\) is very likely the largest one.
=== Specific numbers ===
The apocalyptic number is particularly interesting, being the smallest one containing 666 twice:
It also has the first set of five consecutive sixes.
There are two apocalyptic numbers which are also apocalypse numbers (namely, and ).
\(2^{11,666}\) and \(2^{26,667}\) 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 ===
The first apocalyptic number in the form \(^n2\) is \(^52\). Since the last digits of \(^n2\) converge, a 666 will freeze at some point into the convergent digits. In fact one does so at \(n = 1,213\), and there is a finite (but unknown) number of non-apocalyptic numbers of this form. If \(2^{29,784}\) really is the largest non-apocalyptic power of 2, then \(^42\) is the largest non-apocalyptic power tower of 2.
=== Sources ===
===See also===
*Beast number
Category:Numbers
Category:Series
Category:Powers of 2