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Topic: unitary (Egyptian) fractions
Replies: 17   Last Post: Mar 31, 2000 6:38 AM

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Milo Gardner

Posts: 1,105
Registered: 12/3/04
Unitary (Egyptian) fractions
Posted: Mar 31, 2000 6:38 AM
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Tapio's two part question, how do you write 4/733 and 183/733
as 3-term series, is easily answered in terms of

4/733 = 1/184 + 1/67436 + 1/134872

which means that

732/733 = 183/184 + 183/67436 + 183/134872

still a very difficult situation.


3/733 = 1/367 + 1/733 + 1/269011

also implies that

732/733 = 244/367 + 244/733 + 244/269011,

another very difficult problem to write in fewer than 7-terms.

What I was referencing was that an additive selection of
n/733 members be considered to find the 732/733 'optimal'
series,

such as,

367/733 = 1/2 + 1/1466
184/733 = 1/4 + 1/1466 + 1/2932

or,

551/733 = 1/2 + 1/4 + 1/733 + 1/2932

leaving

181/733 = 1/a + 1/b

which I have not been able to find.


Yes, this is a very difficult problem is the shortest series
is the only goal. However, considering the smallest last term
issue, the problem even becomes more interesting.


Regards to all,


Milo














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