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It is known that the Erdos-Straus conjecture is about writing $4/n$ as three unit fractions. My question is whether it is known that if $a>4$ $$ \frac an=\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_k} $$ where $k<a$? Or it is a conjecture similar to Erdos-Straus one with the same hardness? For instance, is it known whether $$ \frac 5n=\frac1{x_1}+\frac1{x_2}+\frac1{x_3}+\frac1{x_4} ? $$

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There is a conjecture for 5/n identical to the Erdos-Straus conjecture and even for k/n, k any positive integer, see the Wikipedia article on the Erdos-Straus conjecture for details. Frank Okoh

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This is a very partial answer. If $n$ is a practical number, then for every $a<n$ there is a reduction of $a/n$ as $$\frac an=\frac1{x_1}+\dots\frac1{x_k}$$ with $k<a$.

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