New examples of the representation of 1 by the sum of reciprocals of semiprime numbers

TL;DR

This paper presents 17 new examples of representing 1 as the sum of reciprocals of semiprime numbers, reducing the number of terms needed compared to previous known examples, and suggests that the minimal number of terms is 47.

Contribution

The paper introduces 17 new minimal examples with 47 terms for representing 1 as a sum of reciprocals of semiprimes, improving upon prior examples with 48 terms.

Findings

17 new examples with 47 terms each

Minimum number of terms likely 47

No examples with fewer than 47 terms found

Abstract

In 1978, Allan.Wm.Johnson obtained an example of the representation of 1 by the sum of reciprocals of the product of two distinct prime numbers. His example has 48 terms [1], and we had no examples which have less than 48 terms until now. In this paper, we construct 17 new examples that have less than 48 terms. Since all of the new examples have 47 terms and we have no examples which have less than 47 terms, it is assumed that the minimum number of terms is 47.