UNIT FRACTIONS AND THE ERDÖS-STRAUS CONJECTURE

This note considers some aspects of finite sums of unit fractions, including associated recurrence relations and conjectures in the context of experimental mathematics. Unit fractions provide a unifying theme. This paper considers conjectures in general and their mathematical context with particular applications to the Erdös-Straus conjecture with unit fractions as the unifying theme in the context of experimental mathematics.


INTRODUCTION
As an exercise in experimental mathematics [2], this note aims to elaborate some conjectures related to Egyptian fractions and harmonic numbers and to explore them with some recurrence relations and continued fractions. In the context of teaching and learning in general they implicitly involve the relatively neglected educational concepts of functional literacy and numeracy [27]. Conjectures have an inherent fascination and challenge because we can neither prove them nor find counter examples [10]. They can encourage non-standard mathematical skills such as shrewd guessing (or conjecturing) [21], considering integer structure [16], and new approaches to viewing the Cartesian plane [8] in the context of the history of mathematical conjectures [9].

EGYPTIAN FRACTIONS
Egyptian fractions are finite sums of distinct unit fractions. Unit fractions are rational numbers with unit numerators and positive integers as denominators. For example, 1/2 + 1/3 +1/6 qualifies as an Egyptian fraction. We shall also consider briefly such non-standard properties as repeated fractions or negative integers [23].
Every positive rational number can be represented by an Egyptian fraction, so that this was one way the ancient Egyptians were able to use these sums as notation for rational numbers [12]. For instance, Fibonacci went further by suggesting what is now referred to as a 'greedy algorithm' for calculating Egyptian fractions [11]. A greedy algorithm seeks locally optimal solutions to a problem at each stage of a problem solving heuristic [6]. In the example, the iterations terminate with finite a expression starting with in which    represents the ceiling function. For instance, 6/17 = 1/3 + 1/51.
The Erdös-Straus conjecture [1] is a related unsolved problem. It can be stated as: there exist positive integers x, y and z such that for every integer n ≥ 2 [3]. For example, for n = 2, (x,y,z) = (2,2,1); when n = 3, (x,y,z) = (2,2,3). When x,y,z are distinct the solution represents an Egyptian fraction for 4/n [25]. Sometimes there are multiple solutions such as when n = 5, (x,y,z) = (2,4,20) and (2,5,10). This case makes us wonder if there are patterns, and there are for some n, but not for all n [14]. As with other famous conjectures, computational mathematicians have verified the truth of the conjecture for very large values of n, but it has not been proved for all n [cf. 15].
Because of the symmetry of the right hand side of (2.1) it can be rearranged as in which B = x(y + z)/yz is constant for a given (n,x) pair (Table 1) and corresponding (x,y,z) triplets are shown in Table 2 pB is odd and changes by 4 when x changes by 1 ( Table 1), but some pB values do not give integer solutions. Different p values may have the same pB values ( Figure 1). In this case the primes are in the same class. For example, a pB of 1 and 5 (in  [17], whereas a pB of 3 and 7 occurs for  Table 5).    Since odd integers in classes 4 1 and 4 3 extend to infinity, the parallel lines in Figure 1 below will be intersected by horizontal (parallel to the x-axis) lines which represent the primes. While these intersections can yield non-integer x, there will be some integer intersections. Thus all primes will have one or more sets of (x,y,z) triples, as will composites since they are products of primes. That is and so on. For example, for n = 6 (= 2x3), (x,y,z) = (2,8,24) and for n = 3, (x,y,z) = (1,4,12).
To calculate y and z from B it is useful to note that z/y = p or kp (Table 3). Since B is known from the pattern in Table 1, that is,   Table l and parts of Table 4 appears below, followed by a graphical representation which emphasises the consistency of the structures.   A further refinement of the work in this section would be to investigate finite sums of reciprocals of distinct n th primes [13], and we look at some other types of finite sums in the next section.

HARMONIC NUMBERS
Harmonic numbers are sums of the reciprocals of the natural numbers. More precisely, the n th Harmonic number is the sum of the reciprocals of the first n natural numbers:

   H
, so that the components are unit fractions. Moreover, it can be seen that Hn is n times the inverse of the harmonic mean of these natural numbers. More generally, the harmonic mean of n numbers x1, x2,…,xn is given by   v(r,t,m). Other examples of these vectors include ,..., , ) 1 , , ( which opens up a whole world of enumeration mathematics [18].
Generalized harmonic numbers of order r can be represented by

RECURRENCE RELATIONS
The harmonic, geometric and arithmetic means all satisfy first order recurrence relations: , , the first being homogeneous (with common ratio r) and the second non-homogeneous (with common difference d). The generalized harmonic numbers have two recurrence relations, one of each kind: (3.1) a non-homogeneous recurrence relation for n, and (3.2) a homogeneous recurrence relation for r.
Fibonacci's Liber Abaci, mentioned in Section 2, effectively introduced the Fibonacci numbers. These can be defined by the second order homogeneous recurrence relation  and applying this to second order homogeneous linear recurrence relations to get the patterns in Table 6 where    The patterns across rows, down columns (and diagonals) are worth further investigation in the context of partial recurrence relations with analogies from partial difference equations to partial differential equations. We can also connect the ordinary Fibonacci numbers with unit fractions in continued fractions. As an illustration consider the continued fraction expansion of φ. Although it is an algebraic irrational number we can specify its continued fraction expansion accurately and precisely. It is an example of the fact that the continued fraction expansion of an irrational number is unique. We start with  