I would like to have a criterion for an extension of the egyptian-fraction representation (Egyptian fractions have numerator $1$). I allow negative fractions, but the occuring denominators have to be distinct.

For example, $\frac{8}{11}$ , for which I could not find a representation $$\frac{8}{11}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$ with positive distinct integers $a,b,c$ (How can I proof that there is none ?), allows the representation $$\frac{8}{11}=\frac{1}{2}+\frac{1}{4}-\frac{1}{44}$$ which is superior compared to $$\frac{8}{11}=\frac{1}{2}+\frac{1}{6}+\frac{1}{22}+\frac{1}{66}$$ even in two ways : The length is smaller AND the absolute value of the largest denominator is smaller.

Even a fraction like $\frac{36}{457}$ allows the representation $$\frac{1}{13}+\frac{1}{540}-\frac{1}{3208140}$$ The greedy algorithm (always choosing the largest possible fraction) would lead to a long representation with very large denominators.

Does someone know a relatively easy criterion whether a given fraction $\frac{a}{b}$ with positive integers $0<a<b$ and $gcd(a,b)=1$ can be represented as a sum of $k$ egyptian fractions (as said, negative denominators are allowed, but no duplicate absolute values of the denominators) ? I am especially interested in the cases $k=2$ and $k=3$.

  • $\begingroup$ For $k=3$ it means $\frac{a}{b}= 1/A+1/B+1/C = \frac{AB+AC+BC}{ABC}$ $\endgroup$ – reuns Dec 19 '16 at 10:20
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    $\begingroup$ A simple search to wiki shows that for k=3, there exists two conjectures, Erdos-Straus for 4/n and Sierpinski for 5/n, and that both are still undecided to this day. So I doubt there is a general criteria. $\endgroup$ – zwim Dec 19 '16 at 10:21

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