1. The RMP 2/nth table wrote all 2/pq series,
except for 2/95, as exact unit fraction series
by an ancient Egyptian rule, that is equivalent to:
a. 2/pq = 1/a x a/pq
where a = (p + 1) and (p + q).
Interestingly, the selection of a = (p + q) is
a special case that does produced a more concise
series than (p + 1) for the 2/35 and 2/91 cases.
In addition, two other forms may have been
historical known to the ancient scribes, like
Ahmes, as noted by:
b. 2/pq = (1/p + 1/q)2/(p + q)
as seen an an inverse Golden Proportion, 2/p = A x H
the product the arithmetic mean (A) and harmonic mean (H)
in the form of:
b1: 2/pq = 1/A x 1/H
and/or
c. n/pq = 1/pr + 1/qr
where n = 2, and r = (p + q)/n
as the Akhmim P. is cited by Howard Eves (AN INTRODUCTION
TO THE HISTORY OF MATHEMATICS) as equation c. being an
algebraic identity.
Taking the modern British form of the Golden Rule of
fractions, and looking for a confirmation of an ancient
Egyptian context, consider rule a., stated above, from the
RMP and rule d, from the EMLR, stated below:
d. 1/pq = 1/a x a/pq
can be seen as a 'false postion' -like pick a number
type of mental process. Line 1 of the EMLR lists
1/8 = 1/10 + 1/40
There are several available methods, some historical and
others not historical, that can achieve the desired
exact conversion of 1/8th to 1/10 1/40. I suggest that
rule d is appropriate by letting a = 5, is given by:
1/8 = 1/5 x 5/8 = 1/5 x (1/2 + 1/8) = 1/40 + 1/40.
One way to confirm that the EMLR student scribe selected
various values for a paritioning value a, is shows by
1/8 = 1/25 + 1/15 + 1/75 + 1/200
and,
1/16 = 1/50 + 1/30 + 1/150 + 1/400
two 'out of order' EMLR series.
as reversed engineered, by setting a = 25, as
shown by:
1/8 = 1/5 x (1/5 + 1/3 + 1/15 + 1/40)
= 1/5 x (1/5 + 2/5 + 1/40), since 2/5 = 1/3 + 1/15
= 1/5 x (3/5 + 1/40)
= 1/5 x (25/40)
= 1/25 x 25/8
or,
1/pq = 1/a x a/pq
with a = 25.
Thank you all for considering this ancient form
of exact Egyptian fraction arithmetic as a very
old example of the modern Golden Rule of fractions.
Regards,
Milo Gardner
Sacramento, Calif.
Postscript: The 2/95 case simply is 2/19 times 1/5,
where 2/19 was computed by:
e. 2/19 - 1/a = (2a -p)/ap
with a = 12, and the aliquot parts of 12 (6, 4, 3, 2, 1)
used to additively compute 2a - p, or 24 - 19 = 5
mentally by 3 + 2 rather than 4 + 1, or
2/19 = 1/12 + (3 + 2)/(12*19)
= 1/12 + 1/76 + 1/114
as all 2/p RMP series < 2/101 were computed.
Finally, calculating 2/95 by the multiple 1/5
reveals:
2/95 = 1/60 + 1/380 + 1/570
as Wilbur Knorr partially explained in 1982 (Historia
Mathematica) in his article on Egyptian and Greek
unit fraction series.
The RMP 2/101 series in the RMP followed a simple
EMLR example of 1/p = 1/p(1/2 + 1/3 + 1/6), given by:
2/p = 1/p(1/1 + 1/2 + 1/3 + 1/6)
or,
2/101 = 1/101 + 1/202 + 1/303 + 1/606