The Egyptian Fraction: 
The Unit Fraction Equation 
 

Abstract

The Diophantine equation of the title has its roots in the fractional numeration system of dynastic Egypt, dating from the third millennium BC. The problem of writing 1 as the sum of unit fractions first appears explicitly in the famous Ahmous Papyrus (ca. 1500 BC, problem 23 and 24). In AD 1201 Fibonacci (Leonardo of Pisa) proved that every rational number P/Q can be written as such a sum, thus establishing the completeness of the Egyptian system. In the 1890's J. J. Sylvester of John Hopkins gave the first modern proof that our equation in fact has infinitely many solution by examining the sequence 2, 3, 7, 43, 1807, ..., where each term is the product of all preceding terms, plus 1. Each truncation of this infinite sequence provides a solution to the equation under view. But despite a long and distinguished history, it was not until the age of computers that much progress was made regarding the distribution of solution. Only in 1978 did Hungarian researchers Janek and Skula produce the complete list of solution with k less than or equal to 6 terms, while the complete list for k=7 (26 solution) was published by Brenton and Hill in 1988, after preliminary results by Cao, Liu, and Zhang. 

 The study of this and related Egyptian fraction equation received a boost in the 1980's when a group of researchers at Wayne State University discovered applications of this topic to the structure theory of isolated singular points of four-dimensional topological spaces. Singularity theory, especially the study of degenerate points of analytic and algebraic surfaces over the field of complex numbers, provides models for such cosmological structures as black holes, worm holes, and the big bang. These connections between number theory and four-dimensional geometry also led to the development of new themes in the theory of weighted graphs, which in turn led to new examples of perfect groups given by generators and relations associated with the solution n1, ..., nk of the equation under study. 

  

Accomplishments

The Wayne State University Student Research Program has been assisting on this project since 1993. Accomplishments by undergraduate students to date include: 
  • The discovery of a total of 104 solutions to the equation of the title for k=8, as well as many highly unusual solution with 9 or more terms. 
  • The creation of software that will find all solution for k=8 in about 10 months with current hardware, we are also working on vectorized version of our programs that will run on the university's new Cray supercomputer. 
  • Verification for k less than or equal to 8 of a conjecture of Ke and Sun that for each positive k there is at least one set of k primes p1, ..., pk, such that (greek). 
  • The discovery of several new "perfectly weighted graphs," with application to continued fractions and to presentations of groups. 
  • Improvement of the bound on the non-solvability of the Erdos-Moser equation 1n + 2n + ... + (m-1)= mn. No solution exists for m < 4.9 *   10 9321153 

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Last changed: October 31, 1997
E-mail Comments to:    hogan@math.wayne.edu