email@example.com (Dan Hoey) writes: > > 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.
If it can be expressed as n unit fractions it can be expressed as n distinct unit fractions by repeatedly using the transformation 1/(2k-1) + 1/(2k-1) ==> 1/k + 1/k(2k-1)
> By the way, I found that 732/733 has 2771 different seven-term > representations. The largest denominator appears in the > representation (2305193137933140 33397845 4484 45 7 3 2). The > smallest maximum denominator appears in (26388 20524 7330 45 7 3 2).
Damn, you beat me to it. Just to solve this problem, I modified the SmallMultiples method in my EgyptianFraction Mathematica routines (http://www.ics.uci.edu/~eppstein/numth/egypt/Egypt.m) to use dynamic programming instead of brute force, so now it's not hopeless to solve it that way but it's still slow.
I posted a couple days ago the min-max denominator of any representation (7330, with a ten-term representation) and the best denominator for a nine-term representation (8063). I am now also able to find the best eight-term representation:
732/733 = 1/2 + 1/3 + 1/9 + 1/20 + 1/255 + 1/8796 + 1/12461 + 1/13194 -- David Eppstein UC Irvine Dept. of Information & Computer Science firstname.lastname@example.org http://www.ics.uci.edu/~eppstein/