Problem of the Week 848

An Odd Egyptian Puzzle

Since it's ready, this week's PoW goes out a little early. I [Stan] am interested in any solutions you find to #848. (I know a couple.)

It is known that any fraction with an odd denominator is a sum of reciprocals of distinct odd integers. For example,

      35     1      1       1
     ---  =  -  +  --  +  -----   .
     179     7     19     23807

Find such an expression for 3/179.

Background: Such sums (with no odd restriction) are often called Egyptian fractions. One can attempt to use a "greedy" algorithm on this sort of problem. But, despite the fact that, as proved by Fibonacci, the greedy algorithm for unrestricted Egyptian fractions always halts, it is a famous unsolved problem whether the "odd greedy algorithm for Egyptian fractions" always halts. This open problem is discussed in Klee & Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory (MAA). That book, by the way, appeared in a German translation this week (Birkhauser).

© Copyright 1997 Stan Wagon. Reproduced with permission.