Date: Mar 22, 2000 8:31 PM
Author: eppstein@euclid.ics.uci.edu
Subject: Re: unitary (Egyptian) fractions

Bill Daly <bill.daly@tradition-ny.com> writes:

> 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/