For archival purposes we include Milo's 1997 letters and part of a 2005 letter.
September 1997
Hi Scott,
Two URL's come to mind that discuss Egyptian fractions found in the RMP and elsewehere. The first is Kevin Brown's web page:
http://www.seanet.com/~ksbrown/iegypt.htm Kevin discusses the RMP and Akhmim P., as well as related points that should open many eyes (at all ages).Second is the MAA history of math archive, where queries can be made, such as Egyptian Fractions, or a specific document like the RMP, many times finding my papers. I suggest that the EMLR document and its four (4) 1/p and 1/pq rules should be read and understood before attempting the read the more complex RMP 2/nth table. The URL is:
http://forum.swarthmore.edu/epigone/math-history-listAt this point I would like to say that there are several web sites that should be introduced with caution. One is David Eppstein's 10 algorithms for reading Egyptian fractions, NONE of which come close to reading the historical documents.
The modern test for kids and adults alike is to read the historical documents in the context that they were written, and not in any Eurocentric context (such as Greeks like Pythagoras developed number theory). That is to say, Fibonacci's greedy algorithm is only one idea of Sylvester (1890) that reads only four (4) series in the RMP, as heavily used by Eppstein (and others, as a 'strawman').
Note that your 2/q suggestion (as possibly in agreement with my 2/p algorithm): 2/q = 1/a + (2a -q)/aq
reads all but one RMP 2/p series (the except being 2/101, that is read by an EMLR rule.In closing, you may cite/pst this email on your web site to point out that all RMP 2/pq series, but one (2/95, by a multiple of 2/19 from your 2/q rule) are read and computed by:
2/pq = 2/a x a/pq where a = (p + 1) and a = (p + q).Interesting in 500 AD this exact statement of rational numbers as unit fraction series continued, as the Akhmim P. details by several tables. Two interesting series are 4/17 and 8/17 are read by
a = 20/3 (rather than the Egyptian tradition of integers).Regards,
Milo Gardner Retired Military Cryptanlyst, Sacramento, Calif.
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