Express 36/457 as a sum of distinct unit fractions (fractions with numerator 1).

arithmetic/egyptian.fractions New

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Problem

Express 36/457 as a sum of distinct unit fractions (fractions with
numerator 1).  Can it be done with four or fewer?

[ Editor: problem statement reconstructed from the posting. ]

Show solution

> 1/457 + 1/457 + 1/457 etc.

-- Eugene M Whitehouse <74323.2542@CompuServe.COM>

The original poster wants *different* fractions I'm sure.

One way to do this, and it works for *any* fraction, is to use the
"greedy" algorithm; take the *biggest* possible fraction at each stage.

That works for 36/457 as well as any fraction, although it does get
messy:

36/457 = 1/13 + 11/5941

       = 1/13 + 1/541 + 10/3214081

       = 1/13 + 1/541 + 1/321409 + 9/1033034560129

       = 1/13 + 1/541 + 1/321409 + 1/114781617793 + 8/<YECCHH!!>

That's as far as I'm willing to take that mess!  As you can see, it
WILL eventually get there, because those numerators keep diminishing.
But I suspect that the original poster chose 36/457 probably KNOWING
that in this case the numerators diminish ONLY BY ONE at each stage.

I *DID*, however, find a five fraction solution that's not quite so
messy:

36/457 = 1/14 + 1/140 + 1/9140 + 1/18280 + 1/25592

       =  118820/1663480
        +  11882/1663480
        +    182/1663480
        +     91/1663480
        +     65/1663480
          --------------
          131040/1663480

Can anybody do it in four or less?

-- Jim Waters <jwaters@az.com>

[ Editor: no reply to the last question was found in the collected
postings. ]