By Shalosh B. Ekhad and Doron Zeilberger

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(Appeared in Electronic Research Announcement of the Amer. Math. Soc. 3(1997) 78-82.)

Written: May 1, 1997.

Last Update: July 9, 1997.

At this time of writing, we still need human creative geniuses, like John Horton Conway, to DEFINE such marvelous things as surreal numbers and the audioactive sequence 1, 11, 21, 1211, 111221, 312211, .... But given the definition, all the rest can be done by computerkind (with, at present, routine programming still done by humans.) In ref. [C] of the present paper, Conway begs: `Can you find a proof in just a few pages? Please!' (p. 186) If you count pages modulo routine verification and programming, then the present proof is about .1-page-long.

Most importantly, download the Maple package HORTON , without which the present paper makes little sense.

To understand the present paper, you are advised to read first Steve Finch's fascinating essay on Conway's constant..

If you are skeptical, your computer can reproduce the proof by downloading the input file for Cosmo , and after about two weeks (on nice) you should get the output file for Cosmo .

If you want to construct the periodic table ab initio, and at the same time find how each atom splits after it is acted on by Conway's audioactive operator, the relative abundance of each element, the minimal polynomial for Conway's constant lambda, and its value (to 50 digits), all you have to do (assuming that you have maple and HORTON) is run input file for PTlam , and after less than half an hour, you should get the output file for PTlam .

Added Sept. 26, 2007: Noam Zeilberger kindly told me about Kevin Watkins's interesting paper .

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