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

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 eppstein@euclid.ics.uci.edu Posts: 128 Registered: 12/8/04
Re: unitary (Egyptian) fractions
Posted: Mar 22, 2000 8:31 PM

> 26/1249 =
> 1/49 +
> 1/2449 +
> 1/6245053 +
> 1/40696362770053 +
> 1/1731475485562249549506926522 +
> 1/1558019571407872739990323772114944573152396986805272178 +
> 1/98934242784399918989385559529298980395177208662134783303
> No doubt this can be improved.

Uh, 26/1249 =
1/50 + 1/1249 + 1/62450
1/50 + 1/1225 + 1/3060050
1/49 + 1/2498 + 1/122402
1/49 + 1/2450 + 1/3060050

> 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?

Allan Swett has tested everything up to 10^12.

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

O(sqrt(log b)).
M. Vose. Egyptian fractions. Bull. Lond. Math. Soc. 17, 1985, p. 21.
--
David Eppstein UC Irvine Dept. of Information & Computer Science
eppstein@ics.uci.edu http://www.ics.uci.edu/~eppstein/