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| ``Re: Unit Fractions''
by milogardner on 2006-03-28 10:56:46 |
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| Ahmes additional methods, as implied by his 2/nth
table, shows that 2/pq series were written by three
basic techniques. The most popular technique took the
vulgar fraction n/pq and looked for a first partition
value A that equaled (p + 1) such that:
2/pq = (2/A) x (A/pq)
was used.
For example, let 2/pq = 2/21, or p = 3, q = 7 such that
2/21 = (2/(3 + 1))x (3 + 1)/21
= (1/2) x (1/7 + 1/21)
= 1/14 + 1/42
The second most popular conversion technique was used in
the 2/35 and 2/91 cases, where a form closely related
to Howard Eves' 1961 suggestion:
n/pq = 1/pr + 1/qr,
where r = (p + q)/n
was used.
Filling in the p = 5, q = 7, for 2/35 and
p = 7, q = 13, for 2/91, obtains the series
listed by Ahmes.
The final method, noted in 2/95, factored 95
into 5 x 19, with the following taking place:
2/95 = 1/5 x 2/19
with 2/19 being computed by the Hultsch-Bruins rule,
or A = 12, such that
2/19 -1/12 = (24 - 19)/(12*19)
and finding (3 + 2)/(12*19) = 1/76 + 1/114, or
2/95 = 1/5 x (1/12 + 1/76 + 1/114)
= 1/60 + 1/380 + 1/570
as Ahmes listed.
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