# Conway's Cosmological Theorem

## Richard A. Litherland, Mathematics Department, Louisiana State University

The rule for the sequence

1, 11, 21, 1211, 111221, 312211, 13112221, …

is that each term describes the previous one. Start with 1. This can
be described as "one one", which can be represented by the digits
11. Now what we have is "two ones", or in digits 21. This in turn is
"one two, one one", or 1211. Hence the next term is 1113213211. Of
course, one can generate other sequences by the same rule, using
different starting points. This generation rule was introduced in 1987
by John Conway, who called it "audioactive decay"; it is also known as
"look and say".
Henry Bottomley's page (see link above) about this sequence contains several links to
more information. Conway proved some strange and interesting things
about these sequences, of which the most impressive was the
Cosmological Theorem. The two proofs of this, one by Conway and
Richard Parker, and one by Mike Guy (which gave more information),
were lost. In 1997, Ekhad and Zeilberger gave a proof that relies on
computer verification. It was written for Maple, which I'm not
familiar with, so I decided to write my own (in C). In fact I ended up
with two programs, both of which verify Guy's sharper version of the
theorem, and so may be of interest to someone. In the following links you can download:

- an article (Conway's Cosmological Theorem) dicussing the problem and my proofs of the theorem, in PDF format;
- a gzipped tar archive containing the source code for the programs and some related ones, documentation, and the above article;
- just the documentation (The audioactive package), in PDF format.

## References

**1.** J.H. Conway, * The weird and wonderful chemistry of
audioactive decay*, in: * Open Problems in Communication and
Computation*, T.M. Cover and B. Gopinath,
eds., Springer, 1987, pp. 173–188.

**2.** Shalosh B. Ekhad and Doron Zeilberger,
* Proof of Conway's Lost Cosmological Theorem*,
Electron. Res. Announc. Amer. Math. Soc.
**3** (1997),
78–82.