Revisiting Unit Fractions That Sum To 1

TL;DR

This paper investigates the maximum length of sequences of distinct unit fractions greater than 1/100 summing to 1, identifying 27 solutions with 42 terms and proving no solutions with 43 terms exist.

Contribution

It extends previous work by enumerating all 42-term solutions and using prime analysis to establish the non-existence of longer solutions.

Findings

27 solutions with 42 terms exist

No solutions with 43 or more terms exist

Prime analysis helps determine solution limits

Abstract

This paper is a continuation of a previous paper. Here, as there, we examine the problem of finding the maximum number of terms in a partial sequence of distinct unit fractions larger than 1/100 that sums to 1. In the previous paper, we found that the maximum number of terms is 42 and introduced a method for showing that. In this paper, we demonstrate that there are 27 possible solutions with 42 terms, and discuss how primes show that no 43-term solution exists.