A unit fraction representation of a rational number
is a finite sum of reciprocals of positive integers that equals
. Of
particular interest is the case when the denominators in the representation
are distinct, resulting in an Egyptian fraction representation of
.
Common algorithms for computing Egyptian fraction representations of a given
rational number tend to result in extremely large denominators and cannot be
adapted to restrictions on the allowed denominators. We describe an algorithm for
finding all unit fraction representations of a given rational number using
denominators from a given finite multiset of positive integers. The freely available
algorithm, implemented in Scheme and available on GitHub, is particularly well
suited to computing dense Egyptian fraction representations, where the allowed
denominators have a prescribed maximum.
Keywords
Egyptian fractions, unit fractions, number theory,
algorithms, implementation