```Date: Mar 20, 2000 9:04 PM
Author: Dan Hoey
Subject: Re: unitary (Egyptian) fractions

Hugo van der Sanden wrote:> Prompted by a recent reference to the section (D11) on Egyptian> fractions in Guy's 'Unsolved Problems in Number Theory', I noticed> mention of this problem (paraphrased):>   Order the rationals a/b, 0 < a/b < 1, (a, b) = 1, by increasing a+b> and then by increasing a: 1/2, 1/3, 1/4, 2/3, 1/5, 1/6, 2/5, 3/4 ...>   What is the earliest such rational that cannot be expressed as the> sum of n unit fractions?I'm sure you mean n _distinct_ unit fractions.> I was surprised that so few results were known: 2/3, 4/5, 8/11, 16/17,> 77/79 cannot be expressed as the sum of 1, 2, 3, 4, 5 unit fractions> respectively. So I write a simplistic perl program, which has found> (after just over a mips-year of work) that 732/733 is the first in the> list that cannot be expressed as the sum of 6 unit fractions.> Since this program exercises perl beyond the limits of its integer> accuracy, I cannot guarantee the result - can anyone confirm it for me?I dusted off some old Egyptian fraction code I wrote a couple of yearsago.  It uses Common Lisp's exact integer arithmetic, so roundoff isnot an issue.  I verified that all fractions up to 732/733 havesix-term representations and that 732/733 requires seven terms.It took about 13 hours on a 296 Mhz Ultra.  If a mips is aMhz, that's almost half a mips-year.  I don't how much of theremaining speed difference is due to the programming language/systemand how much is due to the algorithm.By the way, I found that 732/733 has 2771 different seven-termrepresentations.  The largest denominator appears in therepresentation (2305193137933140 33397845 4484 45 7 3 2).  Thesmallest maximum denominator appears in (26388 20524 7330 45 7 3 2).Dan Hoey <haoyuep@aol.com>               Posted and e-mailed
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