An Egyptian fraction expansion is a sum of reciprocals of integers, for example:


$$\frac{4}{17} = \frac{1}{5} + \frac{1}{29} + \frac{1}{1233} + \frac{1}{3039345}$$


Every positive rational number $p/q$ has such an expansion, although it is not unique:


$$\frac{4}{17} = \frac{1}{5} + \frac{1}{30} + \frac{1}{510}$$


Let $\ell(p/q)$ = the number of terms in a minimal (i.e. fewest terms) Egyptian fraction expansion of $p/q$.


Given a positive rational $p/q$, how can we compute $\ell(p/q)$?



What is the shortest length of an Egyptian fraction expansion for a given $p/q$?
If there is an expansion with $k$ terms, one of the denominators is at most $kq/p$. So to check whether there is an expansion with at most $k$ terms: for each $m$ from $\lceil q/p \rceil$ to $\lfloor kq/p \rfloor$, check recursively whether $p/q - 1/m$ has an expansion with at most $k-1$ terms.


Whether there is a polynomial-time algorithm is, I think, an open question. 


