I would like to take this opportunity to thank the list owner for implementing a long overdue list. To begin at the beginning, I would like to propose the oldest known descrete mathematics, at least within the written record of the Western Tradition.
Exactly writing a rational number, p/q, as a unit fraction series was well known in the Egyptian Old Kingdom and plausibly the oldest discrete mathematics. We have only fragments of Old Kingdom texts, however, a Horus-Eye doubling methodology, also used for Egyptian multiplicaiton, is well described in several Middle Kingdom texts. The most famous is the Ahmes Papyrus, or as many prefer to title it, the Rhind Mathematical Papyrus (RMP).
The RMP begins with two tables, the 2/n and n/10, all exact unit fraction series. I will describe the 2/nth table, as built upon another Middle Kingdom text, a series of 26 1/p and 1/pq series found in Egyptian Mathematical Leather Roll,EMLR, also purchased by Henry Rhind around 1858 in Egypt.
The EMLR used a simple 1/pq conversion rule 11 times, a rule that was used in the RMP 2/nth table 24 times. Line 1.0 of the EMLR stated:
1/8 = 1/10 + 1/40
an exact series that could have been created by several methods. I prefer to offer:
1/pq = 1/A x A/pq
as the historical series, following A = 5 and A = 25
1/8 = 1/5 x 5/8
= 1/5 x (1/2 + 1/8)
= 1/10 + 1/40
as also used in the RMP with A = (p + 1) and (p + q)
The RMP A = (p + 1) case was used in 22 of 24 series, with A = (p + q) for 2/35 and 2/91.
The one 2/pq series that did not use this common EMLR and RMP rule was 2/95. The series was computed by 2/19 x 1/5 with 2/19 being computed by a 2/p rule.
The RMP 2/p rule was first re-discoverd by Hultsch in 1895, a method that used the aliquot parts of the first partition A, following the algebraic identity rule:
2/p - 1/A = (2A -p)/Ap
with A = 12 being mentally selected (by Ahmes and/or by someone that created the table 200 years earlier).
Re-stating the 2/p rule, in the form,
2/p = 1/A + (2A -p)/Ap
and analyzing the 2/19 case, with A = 12, consider:
2/19 = 1/12 + (24 - 19)/(12*19)
with 2A - p = 5 being computed by two alternative sets of divisors of 12. The two are (3 + 2) and (4 + 1), with Ahmes selecting ( 3 + 2), or
2/19 = 1/12 + (3 + 2)/(12*19)
= 1/12 + 1/76 + 1/114
(which, of course allows 2/95 = 1/5 x 2/19 = 1/60 + 380 + 1/570, as the RMP lists).
Interestingly, Ahmes always selects largest last term series for from the available set of aliquot parts. Note also that no more than three aliquot parts of A were used in the 2/nth table.
Finally, the last 2/p series to be explain in the RMP is 2/101. This 'odd' RMP series is not odd for the EMLR, having been used four times (as I recall) following the simple rule:
2/p = 1/p x (1 + 1/2 + 1/3 + 1/6)
for 2/101 = 1/101 x (1 + 1/2 + 1/3 + 1/6)
= 1/101 + 1/202 + 1/303 + 1/606
Again, I thank the list owner for this opportunity. All comments, pro and con, will be graciously received, and commented on - publically or privatively (depending upon your wishes).