Subject: [HM] Egyptian fractions work of Mahavira
From: Milo Gardner (milo.gardner@24stex.com)
Date: Thu Dec 23 1999 - 11:41:30 EST
Dear HM list members:
Does any recall the 850 AD Egyptian fractions work of Mahavira? R.C. Gupta,
writing in the HPM Newsletter #29, July 1993, edited by Victor Katz, detailed
the following parametric method:
Let r = (q + x)/p, meaning p divides q + x, exactly, such that:
p/q = 1/r + x/qr
letting x = 1, 2, ..., as needed.
For example, Gupta cited 2/5 from the RMP with p = 2, q = 5, x = 1,
and r = 6/2 = 3, or
2/5 = 1/3 + 1/15
Gupta titled this method in the names of Mahavira-Fibonacci, since Fibonacci
later, writing in Liber Abaci, used a related parametric method (as posted
here on HM earlier this year).
I would like to also point out in the x = 1 case
r = q + 1
dominates the RMP, computing all 2/pq series but 2/35, 2/91 and 2/95, as I
have reported in the form of
2/pq = 2/A + A/pq
with A = (p + 1) and (p + q).
To compute 2/35, and 2/95: x = 25 is required and for 2/91: x = 49 is required
to begin BUT not complete the historical series, in the 2/95 case.
To totally compute the RMP 2/nth table two early forms of algebraic identities
like:
2/pq = 2/(p + 1) + (p + 1)/pq
and the Hultsch-Bruin's method,
2/p = 1/A + (2A - p)/Ap,
that may have been known to Ahmes, with A, a highly composite number being
selected from the range:
p/2 < A < p
with the largest last term set of divisors of A, that add up to (2A - p),
required to be found to complete the method.
Note also that consideration of the 2/101 method reported in the EMLR, is
required to fully analyze the RMP 2/nth table.
As an additional RMP example, as Gupta only introduced, but did not compute,
2/43 = 1/42 + 1/86 + 1/129 + 1/301
would have required the Mahavira-Fibonacci method to select x = 41, a task
not easily done in one's head (as the Hultsch-Bruins method may have been
understood by Ahmes).
That is, would anyone like to discuss other aspects of this under reported
(but much discussed in fragmented ways) the historical issue of Egyptian
fractions? I would like to suggest the use of Occam's Razor, such that the
easiest method is reported as the historical one.
Given that Ahmes's 2/nth table probably was not a series of guesses, and with
is clear set of patterns, highly unlikely, does anyone agree with me that
Egyptian fractions should begin to be reported as the earliest form of number
theory, or the earliest form of finite mathematics?
Regards to all,
Milo Gardner
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