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Topic:
A historical 28/97 Egyptian fraction series
Replies:
4
Last Post:
May 3, 2000 8:26 AM




Re: A historical 28/97 Egyptian fraction series
Posted:
May 2, 2000 2:23 AM


milogardner@juno.com (milo gardner) writes: > With scanning the book can anyone come up with the unit fractions > series that only used denominators < 1000?
You won't be happy  it requires five terms:
28/97 = 1/4 + 1/40 + 1/97 + 1/485 + 1/776
There are other fiveterm solutions as well:
28/97 = 1/4 + 1/30 + 1/388 + 1/582 + 1/970 28/97 = 1/5 + 1/12 + 1/388 + 1/582 + 1/970
Of course you could get more than five terms, e.g. by expanding the terms with smaller denominators... Here are the best three and four term representations (there is no two term representation):
28/97 = 1/4 + 1/36 + 1/97 + 1/1746 28/97 = 1/4 + 1/26 + 1/5044
> Tomorrow I'll post Bunch's series, and ask: how did the Egyptian > scribe come up with the series.
No idea, but my software found it by trying combinations of multiples of 1/97 until it found one for which the remaining fraction after subtracting those multiples no longer had a factor of 97 in the denominator. The detailed algorithm involves modular arithmetic and dynamic programming, neither of with were likely known in that form to the scribes...
> Since Bunch suggested that the Egyptian fraction method was awkward > modern mathematics should have little intellectual difficulty with > the problem.
Certainly none of these calculations took much computer time.  David Eppstein UC Irvine Dept. of Information & Computer Science eppstein@ics.uci.edu http://www.ics.uci.edu/~eppstein/



