NetNews Usenet Archive 1992 #31/NN_1992_31.iso/spool/sci/math/17375

Text File | 1992-12-23 | 1.7 KB | 53 lines

Newsgroups: sci.math
Path: sparky!uunet!think.com!mintaka.lcs.mit.edu!zurich.ai.mit.edu!ara
From: [e-mail] (Allan Adler)
Subject: Re: sums of reciprocals
Message-ID: <[e-mail]>
Sender: [e-mail]
Organization: M.I.T. Artificial Intelligence Lab.
Distribution: sci
Date: Wed, 23 Dec 1992 19:03:40 GMT
Lines: 41


In an earlier posting, I wrote:

>
>Let f(n)=1/n for every positive integer n. Let x be a positive
>real number. Let g(n,x) be defined as follows:
>
>(1) g(n,x) is a strictly increasing sequence of positive integers, possibly
>     a terminating sequence.
>(2) for all n, g(n,x) is the smallest integer consistent with (1)
>    such that the sum of the reciprocals of g(k,x) for k=1,...,n
>    does not exceed x.
>
>I have been told that if x is rational then this leads to a way of writing
>x as a finite sum of reciprocals of distinct positive integers. Let the number
>of terms in this sum be N=N(x). I have been told that N is also the number of
>terms in a minimal representation of x as a sum of reciprocals of positive
>integers allowing repetitions.
>
>Is this true and if so how or where is it proved?
>

Dan Asimov kindly provided me with an example of a fraction x for which
N=N(x) is not the minimal number of terms in a representation of
x as a sum of distinct reciprocals of  positive integers. Namely,


17/72 = 1/5 + 1/28 + 1/2520

using the algorithm above, but there is a shorter representation,
namely

17/72 = 1/8 + 1/9

The next question is this: let M=M(x) be the minimal number of terms in
a representation of x as a sum of distinct reciprocals of positive
integers. Is N(x)/M(x) bounded?


Allan Adler
[e-mail]