Can you represent 36/457 as a sum of different fractions with numerator 1?

arithmetic/egyptian.fraction New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

Can you represent 36/457 as a sum of different fractions with
numerator 1?

-- Andrew Nikitin <nsg@barcel.kharkov.ua>

Show solution

Here is a shorter answer.  First,   36/457 = 1/13 + 11/(13*457)
The decomposition   11/13 = 1/2 + 1/3 + 1/78   is easy to find.  This
gives

36/457 = 1/13 + 1/914 + 1/1371 + 1/35646

Improvements, anyone?

-- Barry Wolk <wolk@ccm.umanitoba.ca>

I'm not sure what constitutes an improvement, but here's one with the same
number of denominator digits but a larger fourth term:

 36/457 = 1/13 + 1/611 + 1/5941 + 1/21479

and one with fewer denominator digits but a smaller fourth term.

 36/457 = 1/16 + 1/71 + 1/457 + 1/519152

I haven't yet found one that's unambiguously an improvement, though.

-- Jim Gillogly <jim@acm.org>