Properties of Egyptian Fraction Expansions for Rational Numbers Between 0 and 1 Obtained by Using Engel Series

The ancient Egyptians expressed rational numbers as the finite sum of distinct unit fractions.  Were the Egyptians limited by this notation?  In fact, they were not as every rational number can be written as as a finite sum of distinct unit fractions.  Moreover, these expansions are not unique.  There exist several different algorithms for computing Egyptian fraction expansions, all of which produce different representations of the same rational number.  One such algorithm, called an Engel series, produces a finite increasing sequence of integers for every rational number.  This sequence is then used to obtain an Egyptian fraction expansion.  Motivated by the work of M. Mays, this project aims to investigate properties of natural number denominators n, that produce length x Egyptian fraction expansions using Engel series for x/n between 0 and 1.  While computing Engel expansions using Mathematica, a helpful pattern emerged.  We noticed that rational numbers x/n which produce length x Egyptian fraction expansions were those whose denominator n was divisible by every natural number between x and 2.  We conjecture that this will always hold for length x Engel series of the form x/n between 0 and 1.  Furthermore, we conjecture that an algorithm for finding the smallest n such that x/n produces an x length Engel series is to add one to the lowest common multiple of x, x-1, ..., 3, and 2.  Using properties of the ceiling and floor functions, we were able to verify this holds for some small values of x.