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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. 2157 is the smallest apocalyptic number:

182,687,704,666,362,864,775,460,604,089,535,377,456,991,567,872

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 3,716 non-apocalyptic numbers of the form 2n 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.

## Contents

### Specific numbers

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

1,684,996,666,696,914,987,166,688,442,938,726,917,102,321,526,408,785,780,068,975,640,576

It also has the first set of five consecutive sixes.

There are two apocalyptic numbers which are also apocalypse numbers (namely, 22,210 and 22,212).

$$2^{11,666}$$ 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 almost surely 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.

### See also

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