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The ancient Egyptians expressed rational numbers as the finite sum of distinct unit fractions. Were Theorem 1. Suppose x is a positive integer and n = lcm(x, x 1,...,3, 2) + 1. Then x will have − n the Egyptians limited by this notation? In fact, they were not as every rational number can be written a length x Engel expansion. Through this project we have shown a method for choosing a denominator that will always produce as a finite sum of distinct unit fractions. Moreover, these expansions are not unique. There exist an Engel expansion whose length is equal to its numerator. We have also demonstrated that while this is not the only possible way to produce a length x expansion, n = lcm(x, x 1,...,3, 2) + 1 is several different algorithms for computing Egyptian fraction expansions, all of which produce different − conjectured to be the least denominator that does so for some fixed x. Tables 1 and 2 demonstrate representations of the same rational number. Example 2. Let x =4. Then by Theorem 1, n = lcm(4, 3, 2) + 1 = 13. In order clarify why One such algorithm, called an Engel series, produces a finite increasing sequence of integers for every this for small values of x. There are other values for n which produce a length x Engel expansion this n produces the desired expansion, 13 will be written as lcm(4, 3, 2) + 1 for the majority of this x rational number. This sequence is then used to obtain an Egyptian fraction expansion. Motivated by 4 for n. example. To find the Engel expansion, first let u1 = . The first term of the Engel series the work of M. Mays, this project aims to investigate properties of natural number denominators n lcm(4,3,2)+1 The paper includes more extensive tables for small numer- x will be 3 ators that demonstrate that n = lcm(x, x 1,...,3, 2)+1 is that produce length x Egyptian fraction expansions using Engel series for n between 0 and 1. 1 lcm(4, 3, 2) + 1 lcm(4, 3, 2) 1 Engel expansions for n − a1 = = = + . the least n that will produce an a length x Engel expansion While computing Engel expansions using Mathematica, a helpful pattern emerged: rational numbers "u # " 4 # " 4 4# x k 1 x 1 x which produce length x Egyptian fraction expansions were those whose denominator n = lcm(x, x n i=1 a1 a2 ai for . We are open to the possibility that a counter exam- n lcm(4,3,2) · ··· n − 3 1 1 x 1,...,3, 2) was divisible by every natural number between (and including) x and 2. We conjecture By the lemma, since 4 =3is an integer,a1 can be rewritten as: + ple may be found. That is, some n$