Sylvester's Conjecture and the Egyptian Fractions

TL;DR

This paper explores Sylvester's conjecture by developing operators to generate Egyptian fractions of 1 and identifying patterns that produce new, coprime numbers, aiming to advance understanding of Egyptian fractions.

Contribution

It introduces a novel set of operators for generating Egyptian fractions and a pattern detection method to find coprime numbers, advancing the study of Sylvester's conjecture.

Findings

Operators can generate all Egyptian fractions of 1

Patterns reliably produce coprime new numbers

Potential pathway to prove Sylvester's conjecture

Abstract

This paper attempts to prove the Sylvester's conjecture using Egyptian Fractions with two key ingredients. First, creating a set of operators that completely generates all possible Egyptian fraction of 1. And second, to detect patterns in every operator that surely will generate a new number which are relatively prime to all that came before.