I recently read about the study of Egyptian Fractions on the Good Math, Bad Math blog. The references to this article show that many years of research have gone into trying to find efficient ways to calculate the minimum length forms.
Are there any real-world applications that computing minimum-length Egyptian forms can be applied to?
Electronic circuit design is one area where Egyptian fractions have practical use. When electrical resistors are added to a parallel circuit, the reciprocals are added to find the total resistance. An example is a parallel circuit with 1,2,3, and 6 Ohm resistors.
$$\frac{2}{1\Omega}=\frac{1}{1\Omega}+\frac{1}{2\Omega}+\frac{1}{3\Omega}+\frac{1}{6\Omega}$$
Inductance (L) in Henrys (H) in parallel circuits is calculated in the same way.
$$\frac{1}{2 H}=\frac{1}{4 H}+\frac{1}{5 H}+\frac{1}{36 H}+\frac{1}{45 H}$$