Heinz recently cited one of seven Fibonacci methods
to convert rational numbers to unit fraction series
(other than his famous greedy algorithm). Heinz
cited the equivalancy of:
n/p - 1/a = (n*a - p)/(a*p)
or,
n/p = 1/a + (n*a - p)/(a*p) (identity 1.0)
where a = p + 1, or higher
or,
n/p = 1/(p + 1) + n*(p + 1)/((p + 1)*p)
(as I recall. Please excuse any error listed above)
Looking at identity 1.0 by induction, it is clear that
I. 1/p = 1/(p + 1) + 1/(p + 1)/((p + 1)*p)
works for any p > 1, such as p 2, 3, ...
and,
II 2/p = 1/(p + 1) + 2/(p + 1)/((p + 1)*p)
also works for any p > 1
and so forth, as Fibonacci suggested, and Heinz implied
can be proven by induction.
My question is, can Heinz or someone else list the other
six (6) methods, other than the greedy algorithm, that
Fibonacci used to exact convert any rational number to
a short and concise unit fraction series, and offer
a suggestion why several methods were practically needed
(beyond a mathematician's usual probing mind)?
I would be greatly interested in comparing this set of
methods to earlier Greek and Egypt methods that are
implied by the Akhmim, Hibeh, EMLR, RMP 2/nth table,
and the Moscow Papyrus, dating back to 2,000 BC, that
have often be read only as practical in nature.
Regards to all,
Milo Gardner