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From: ksoileau on 5 Feb 2010 15:05
Proof or counterexample, please:

Every rational between 0 and 1 can be written as a sum of reciprocals
of distinct positive integers. If the fractions are sorted in
increasing denominator order, this representation is unique. For
example: 131/483=1/3+1/7+1/483.

DISCLAIMER: I am not a student, just interested in this question.
Thanks to all...

Kerry Soileau
From: Maarten Bergvelt on 5 Feb 2010 15:16
On 2010-02-05, ksoileau <kmsoileau(a)gmail.com> wrote:
> Proof or counterexample, please:
>
> Every rational between 0 and 1 can be written as a sum of reciprocals
> of distinct positive integers. If the fractions are sorted in
> increasing denominator order, this representation is unique. For
> example: 131/483=1/3+1/7+1/483.

What about 2/9?

--
Maarten Bergvelt
From: Arturo Magidin on 5 Feb 2010 15:28
On Feb 5, 2:05 pm, ksoileau <kmsoil...(a)gmail.com> wrote:
> Proof or counterexample, please:
>
> Every rational between 0 and 1 can be written as a sum of reciprocals
> of distinct positive integers. If the fractions are sorted in
> increasing denominator order, this representation is unique. For
> example: 131/483=1/3+1/7+1/483.

Disproof of uniqueness: 1/1 = (1/2) + (1/3) + (1/6).

If you want the rationals strictly between 0 and 1, then

(1/2) = (1/4) + (1/6) + (1/12) = (1/4) + (1/5) + (1/20)

--
Arturo Magidin
From: Arturo Magidin on 5 Feb 2010 15:30
On Feb 5, 2:16 pm, Maarten Bergvelt <be...(a)math.uiuc.edu> wrote:
> On 2010-02-05, ksoileau <kmsoil...(a)gmail.com> wrote:
>
> > Proof or counterexample, please:
>
> > Every rational between 0 and 1 can be written as a sum of reciprocals
> > of distinct positive integers. If the fractions are sorted in
> > increasing denominator order, this representation is unique. For
> > example: 131/483=1/3+1/7+1/483.
>
> What about 2/9?

1/9 + 1/10 + 1/90 = 2/9

More generally, (2/k) = (1/k) + (1/(k+1)) + (1/k(k+1)).

--
Arturo Magidin
From: James Dow Allen on 5 Feb 2010 15:30
On Feb 6, 3:05 am, ksoileau <kmsoil...(a)gmail.com> wrote:
> Proof or counterexample, please:
>
> Every rational between 0 and 1 can be written as a sum of reciprocals
> of distinct positive integers. If the fractions are sorted in
> increasing denominator order, this representation is unique. For
> example: 131/483=1/3+1/7+1/483.

The reciprocals are called "Egyptian fractions." They've been
used since ancient times. Representation is certainly *not* unique,
for example:
17/21 = 1/2 + 1/6 + 1/7 "optimal"
17/21 = 1/2 + 1/4 + 1/17 + 1/1428 "greedy"

I think there's an unresolved conjecture by Erdos that
4/n = 1/a + 1/b + 1/c
has solutions for all n > 1.

<shameless self-promotion>
I found all the above at
http://fabpedigree.com/james/mathmen.htm
</shameless self-promotion>

James Dow Allen
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