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 Yamato 2013-05-12 14:16

An Egyptian Fraction

Write [TEX]\frac{99}{10001}[/TEX] as a sum of distinct unit fractions (all numerators are 1), where all denominators are [B]smaller[/B] than 10001. :smile:

 c10ck3r 2013-05-12 15:01

[SPOILER]1/102+1/10517+1/228264101+1/188377806760266318[/SPOILER]

 CRGreathouse 2013-05-12 16:04

cl10ck3r, that doesn't meet the denominator requirement.

I get [spoiler]99/10001 = 1/105 + 1/6165 + 1/9198 + 1/9590[/spoiler].

 c10ck3r 2013-05-12 16:11

[QUOTE=CRGreathouse;340147]cl10ck3r, that doesn't meet the denominator requirement.

I get [spoiler]99/10001 = 1/105 + 1/6165 + 1/9198 + 1/9590[/spoiler].[/QUOTE]
Ah, glanced over that part.

 Batalov 2013-05-12 19:35

Many solutions. Here's just a few:
[SPOILER]v=[3,14,15];sum(k=1,#v,1/(73*v[k])+1/(137*v[k]))
v=[3,10,42,70];sum(k=1,#v,1/(73*v[k])+1/(137*v[k]))
v=[5,6,14,30];sum(k=1,#v,1/(73*v[k])+1/(137*v[k]))
v=[7,10,14,15,21,35,70];sum(k=1,#v,1/(73*v[k])+1/(137*v[k]))[/SPOILER]

 CRGreathouse 2013-05-12 21:15

[QUOTE=Batalov;340166]Many solutions. Here's just a few:
[SPOILER]v=[3,14,15];sum(k=1,#v,1/(73*v[k])+1/(137*v[k]))
v=[3,10,42,70];sum(k=1,#v,1/(73*v[k])+1/(137*v[k]))
v=[5,6,14,30];sum(k=1,#v,1/(73*v[k])+1/(137*v[k]))
v=[7,10,14,15,21,35,70];sum(k=1,#v,1/(73*v[k])+1/(137*v[k]))[/SPOILER][/QUOTE]

Using [spoiler](99/10001)/(1/73+1/137) = 33/70[/spoiler]. Convenient, but not good if you're trying to minimize terms (as I was). Also interesting would be minimizing the maximum term, maximizing the minimum term, and minimizing the maximum ratio (or difference) between terms.

 Batalov 2013-05-12 21:41

Yes, indeed. There was a P.Euler problem in similar vein. (more than one, actually)

Can we count the number of (distinct) solutions of OP?
(with d[SUB]1[/SUB] < d[SUB]2[/SUB] < ... < d[SUB]n[/SUB] < 10001)

 CRGreathouse 2013-05-13 05:11

Good question!

It seems very hard, unless the upper bound on the number of terms can be lowered substantially from the naive 98. I don't see why it should be, though.

 Yamato 2013-05-16 15:50

[QUOTE=CRGreathouse;340173]Using [spoiler](99/10001)/(1/73+1/137) = 33/70[/spoiler]. Convenient, but not good if you're trying to minimize terms (as I was). Also interesting would be minimizing the maximum term, maximizing the minimum term, and minimizing the maximum ratio (or difference) between terms.[/QUOTE]

Thx for your postet solutions, this is an interesting approach to solve this problem.

The "generic" method (which also works for prime denominators) is to choose a smooth number k which factors into small primes and write the numerator in (99k)/(10001k) as a sum of factors of the denominator. This leads to a unit fraction expansion for 99/10001, e.g. 1/247 + 1/292 + 1/949 + 1/1898 + 1/2603 + 1/3562 + 1/5548.

An interesting question is to how to find the expansion, where all denominators are bounded by a number as small as possible.

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