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

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 Bill Daly Posts: 69 Registered: 12/8/04
Re: unitary (Egyptian) fractions
Posted: Mar 22, 2000 8:12 PM
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For what it's worth, I have tested all proper fractions a/b with b prime
and less than 3000, and all of them have an Egyptian fraction
representation of length 7 or less. For some reason, the most
time-consuming fraction was 26/1249, for which I found the following
representation after a little less than 16 minutes:

26/1249 =
1/49 +
1/2449 +
1/6245053 +
1/40696362770053 +
1/1731475485562249549506926522 +
1/1558019571407872739990323772114944573152396986805272178 +
1/98934242784399918989385559529298980395177208662134783303

No doubt this can be improved.

A couple of questions:

1) What is the smallest N for which it is not known whether 4/N has an
Egyptian fraction representation of length 3?

2) Is it known whether the length of the minimal representation for a/b
is O(log(b)), or something similar?

Regards,

Bill

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