Sam's point that Egypt's political sub-division was named nomos
only hints at a very old mathematical parallel. Nome may have
meant the partition of any rational number, following the Egyptian
fraction tradition, as Egyptians have been documented to have taught
Greeks. I wonder what the Egyptian word was for partition, say
during the Middle Kingdom?
May be the same word used for:
1/p, 2/p and n/p partitions
and for
1/pq, 2/pq and n/pq partitions?
Or could there have been two different words, since two classes
of algebraic identities were used, one based on aliquot parts,
equivalent to nome, and another word for a simpler algebraic
identity method, as outlined by two partitioning, conversion,
rules:
1. n/p = 1/A + (nA -p)/Ap
where n = 2 used in the RMP 2/nth table limited A to a
highly composite number, selected from the range:
p/n < A < p
with A's aliquot parts, A's divisors, being used to
additively compute nA - p, always partitioning n/p,
as Greeks also used in the 300 BC Hibeh Papyrus
n/45 table (for all but one series)
and,
2. n/pq = n/A x A/pq (a simple algebraic identity)
as the EMLR 1/pq tables used A = 5, 25
and,
the RMP 2/pq tables used A = (p + 1), (p + q)
as Greeks continued to use throughout its history,
as found in the Akhmim Papyrus, in the form of
n/pq = 1/pr + 1/qr, where r = (p + q)/n.
Regards to all,
Milo Gardner