I am grateful to Heinz for his list of eight (8) none
greedy algorithms that Fibonacci used to partition
a/b, any rational number, into a concise unit fraction
series. Overall, Fibonacci's 1200 AD methods are
all parametric, and tend to show many of the properties
that Hultsch and others have found in the RMP, and
other ancient Egyptian and Greek unit fraction series.
For example, working backwards from case 8:
1. Choose a natural number k;
2. determine q and r such that ak = qb + r, with r < b
3. Then a/b = q/k + r/(kb)
4. Iterative processes related to case 2 will be explained
later however, it is interesting to note that Heinz's
use of k= 6, 8, 12, 60 with a limitation of
b/2 < k < 2b
follows very closely to first partition properties of
Hultsch's 1895 suggestion of 2/p series, as I read as:
2/p - 1/A = (2A -p)/Ap (identity 1.0)
where,
A was a highly abundant number, such as 6, 8, 12, 60
chosen within the range p/2 < A < p such that the
aliquot parts of a were used to compute (2a -p)/ap;
a method that also always converts 2/p to a unit
fraction series.
As a general method to convert n/p to an Egyptian
fraction series, as Fibonacci has parametrically
suggested, the 300 BC Hibeh Papyrus implied a
modification of identity 1.0 to the form,
n/p - 1/A = (2A -p)/Ap (identity 2.0)
or,
n/p = 1/A + (2A -p)/Ap
as Fibonacci himself may have recognized (as Heinz
lists as case 8).
Case 7, another parametric form:
1. b = aq + r,
2. then 1/(q + 1) < a/b < 1/q
3. and a/b = 1/(q + 1) + (a -r)/(q + 1)b)
The above nearly approximates the composite case that has
forwarded to Heinz for his review. That is, the RMP 2/pq
case:
2/pq = 2/A x A/pq (identity 3.0)
where A = (p + 1), for all but 2/35, 2/91 and 2/95
and A = (p + q) for 2/35 and 2/91
with the 2/95 case being a multiple of the 2/19, computed
by Fibonacci's case 8, appear to be algebraic equivalences.
Note that identity 3.0 can be re-stated various ways, one
being:
2/pq = (1/p + 1/q)2/(p + 1) (identity 3.1)
as more closely related to Fibonacci's above rule (Heinz's
case 7) as shown by letting a - r = 1
then setting a/b = 2/b to directly compare with the RMP
2/b = 1/(q + 1) + 1/(q + 1)b
is an algebraic match, right?
I have gone on too long, so I will not cite Greek texts that
have generalized the composite case of identity 3.0 or 3.1.
That subject Heinz and I can discuss off-line, and share
a summary post, allowing a - r to be any value, at some
future time (as well as the points covering cases 1 - 6,
as posted by Heinz to HM).
I thank everyone for their time.
Regards to all,
Milo Gardner