Computerizing the Andrews-Fraenkel-Sellers Proofs on the Number of m-ary partitions mod m (and doing MUCH more!)

Shalosh B. Ekhad and Doron Zeilberger

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Posted: Nov. 20, 2015.
Last update of this web-page: Dec. 31, 2018.

[Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, as well as in]

Thomas Edison said that genius is one percent inspiration and ninety-nine percent perspiration. Now that we have computers, they can do the perspiration part for us, but we need meta-inspiration, meta-geniuses, and meta-perspiration, to teach the human inspiration to our silicon colleagues. Sooner or later, computers will also do the inspiration part, but let humans enjoy the remaining fifty (or whatever) years left for them, and focus on inspiration, meta-inspiration, and meta-perspiration, and leave the actual perspiration part to their much faster- and much more reliable- machine friends.

In this short article, two recent beautiful proofs of George Andrews, Aviezri Fraenkel, and James Sellers, about the mod m characterization of the number of m-ary partitions are simplified and streamlined, and then generalized to handle many more cases, and prove much deeper theorems, with the help of computers, of course.

Added Dec. 31, 2018 Giedrius Alkauskas informed me that the Andrews-Fraenkel-Sellers theorem should be called the Alkauskas theorem. It was discovered, and first proved (page 9, line 3) in his 1999 Third Course thesis (DOI: 10.13140/RG.2.2.23219.12321), written in Lithuanian.

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Some Input and Output files for the Maple package AFS.txt

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