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EGYPTIAN FRACTION EXPANSIONS for RATIONAL NUMBERS BETWEEN 0 and 1 OBTAINED with ENGEL SERIES 1. Introduction the Rhind Papyrus I

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EGYPTIAN FRACTION EXPANSIONS for RATIONAL NUMBERS BETWEEN 0 and 1 OBTAINED with ENGEL SERIES 1. Introduction the Rhind Papyrus I EGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND 1 OBTAINED WITH ENGEL SERIES ELVIA NIDIA GONZALEZ´ AND JULIA BERGNER, PHD DEPARTMENT OF MATHEMATICS Abstract. 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 a finite sum of distinct unit fractions. Moreover, these expansions are not unique. There exist several di↵erent algorithms for computing Egyptian fraction expansions, all of which produce di↵erent 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 x length x Egyptian fraction expansions using Engel series for n between 0 and 1. x While computing Engel expansions using Mathematica, a helpful pattern emerged: rational numbers n which produce length x Egyptian fraction expansions were those whose denominator minus one was divisible by every natural number between (and including) x and 2. We conjecture and prove that this will always x hold for length x Engel series for between 0 and 1 where n = k lcm(x, x 1,...,3, 2) + 1 for k N. n · − 2 Furthermore, we conjecture that n = lcm(x, x 1,...,3, 2) + 1 is the smallest n such that x produces a − n length x Egyptian fraction expansion. A proof that n = k lcm(x, x 1,...,3, 2) + 1 works is included along · − with an investigation of whether or not n = lcm(x, x 1,...,3, 2) + 1 the least such n that does. − 1. Introduction The Rhind papyrus is responsible for preserving the mathematical methods employed by the ancient Egyptians. It is clear from this document that the Egyptians had an unintuitive way of expressing rational numbers. Unlike the current precedent, in which one integer is written over another integer, they would write 2 1 1 out rational numbers as a sum of distinct unit fractions. For example, 7 would be written as 4 + 28 instead. Additionally, the Rhind papyrus included extensive tables, one of which included di↵erent representations 2 of n for odd n between 5 and 101 using the following identity [7] 1 1 1 = + . n n +1 n (n + 1) · However impractical these expansions may seem, they make certain real-life problems much easier to solve. One such real-life application is the sharing problem. This problem focuses on sharing whole items (like loaves of bread) with multiple people. For example, how can five loaves be split evenly amongst eight 5 people? The solution can be written as the proper fraction, 8 . This means each person gets five eighths of a loaf. Now, applying this is not as easy as writing out the solution as a fraction. How would someone cut five-eighths from eight loaves? Each loaf could be cut in half, then those halves into quarters, then those quarters into eighths. From here, each of the eight persons would receive five small pieces of bread. This solution, although correct, is not very practical. 5 1 1 If we represent 8 as 2 + 8 , this sharing problem becomes significantly easier to put into practice. Each person receives half a loaf first, which ensures that four loaves are distributed evenly amongst all eight people. Key words and phrases. Egyptian fractions, Engel Series, Number Theory. This work is funded through UC LEADS at the University of California, Berkeley. 1 2 ELVIA NIDIA GONZALEZ´ AND JULIA BERGNER, PHD DEPARTMENT OF MATHEMATICS Next, the final loaf can be cut into eighths and each person would receive one of those pieces. Along with being easier to distribute the bread, this solution allows each person to have one larger piece (the half loaf) along with the smaller piece (the eighth loaf). These representations inspired many questions. Some have been answered. For example, we know that Egyptian fraction expansions are not unique. Di↵erent algorithms may produce di↵erent representations 5 1 1 1 of the same fraction. Our example of 8 can also be expressed as 2 + 10 + 40 . Table 1 lists several more examples. Table 1. Expansions of various rational numbers using di↵erent algorithms. x n Engel Series Greedy Algorithm Continued Fraction 5 1 1 1 1 1 1 1 1 1 1 7 2 + 6 + 24 + 168 2 + 5 + 70 2 + 6 + 21 4 1 1 1 1 1 1 1 1 1 1 1 13 4 + 20 + 140 + 1820 4 + 18 + 468 4 + 28 + 70 + 130 7 1 1 1 1 1 1 1 1 18 3 + 18 3 + 18 3 + 24 + 104 + 234 Another question that arose from Egyptian fraction expansions inquired whether or not the Egyptians were limited by the use of this notation. That is, could they express any rational number as a sum of distinct unit fraction? The Egyptians, as it turns out, were not at all limited. Any rational number can be expressed as the finite sum of distinct unity fractions. Furthermore, even irrational numbers can be expressed as an infinite sum of distinct unit fractions. The focus of this project is Egyptian fraction expansions for rational numbers between 0 and 1 (but not including 0 and 1) obtained using an Engel series. Some examples of these expansions can be seen on the second column of Table 1. 2. Engel Expansions An Engel series, for any real number, is sequence of increasing integers [6]. Any real number, say z, can be expressed as an Egyptian fraction expansion using an Engel series in a unique way. This representation for any real z can be computed by defining u = z then letting a = 1 .From here, each subsequent a 1 1 u1 i+1 d e 1 and ui+1 is obtained by first computing ui+1 = ui ai 1 then computing ai+1 = ,where is the · − d ui e de ceiling function defined as r = s if and only if s 1 <r s [2, Page 69, (3.5)]. d e − If z is a rational number, this process will halt whenever uk = 0 and the Engel series is a1,a2,...,ak 1 . The Egyptian fraction expansion for a rational number is derived from the series as follows:{ − } k 1 − 1 1 1 1 z = = + + + . a a a a a a ··· a a a i=1 1 2 i 1 1 2 1 2 k 1 X · ··· ··· − Otherwise, if z is irrational, this algorithm continues and we end up with an infinite Engel series a1,a2,a3,... This infinite Engel series can be used to obtain an Egyptian fraction expansion with the following{ sum [3]: } 1 1 . a a a i=1 1 2 i X · ··· Any Egyptian fraction expansion derived using an Engel series will be referred to as an Engel expansion from now on. Below we prove that Engel expansions are finite for all rational numbers. Theorem 2.1. Engel expansions are finite for all rational numbers. EGYPTIANFRACTIONEXPANSIONSUSINGENGELSERIES 3 x Proof. Suppose m Q.Thenm = n for x, n Z and n = 0. Begin to compute the Engel expansion by first letting 2 2 6 x n u = and a = . 1 n 1 d x e Next, compute u2 using a1 and u1 to obtain x x a n u = u a 1=( a ) 1= · 1 − . 2 1 · 1 − n · 1 − n 1 n n n By the definition of the ceiling function, = = a1 if and only if a1 1 < a1.Thusa1 1 < d u1 e d x e − x − x if and only if x a1 x<n. Subtracting n from both sides of the inequality and adding x to both leaves us with x a n<x.· −The numerator for u is x a n. From the inequalities above, x a n<xwhich is · 1 − 2 · 1 − · 1 − the numerator for u1. We can now find another positive integer a such that a (x a n) n<x a n. The left hand side 2 2 · 1 − − · 1 − of this inequality is precisely the numerator for u3 and the right hand side is precisely the numerator for u2. So no matter which ui we start with, the numerator for ui+1 will be less than the numerator for ui.Since the numerators for the ui’s are strictly decreasing, eventually we will end up with zero in the numerator for some ui (say uk has a numerator of 0) and this process halts with k 1 − 1 a a a i=1 1 2 i X · ··· which is a finite Engel expansion as desired. ⇤ 3. Results The work within this project was inspired by a paper by Michael E. Mays titled, “A Worst Case of the Fibonacci-Sylvester Expansion” [4]. Mays explored Egyptian fraction expansions of rational numbers using the greedy algorithm. More specifically, his paper investigated the properties of fraction expansions which had lengths that matched the numerator using the greedy algorithm. In a similar manner, this paper investigates fraction expansions whose lengths match their numerators using Engel expansions. Initially, this project began by simply looking at tables of Engel expansions. Tables 2 and 3 o↵er a few examples of Engel expansions whose lengths match their numerators. Each table begins with a denominator that is one greater than the numerator and ends when an Engel expansion of length equal to the numerator is obtained. The first column is simply the rational number being investigated, the second column is the Engel series, and the third column is the Engel expansion 1.
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