A Novel Approach in Number Theory for Representing Large Numbers: The Arrow-Free Notation

This article introduces a new notation for expressing extremely large numbers, based on the hyperoperation concept in group theory. The method employs a finite sequence of positive integers separated by specific notational symbols, allowing for concise representation through an arrow-free notation: ( ​), where b represents the number of copies of a, and n denotes the arrow’s number described by a general formula. This recursive definition aims to replace the Knuth up-arrow notation and Conway chained arrow notation, which require the insertion of arrows between or within numbers. The new approach simplifies these expressions, eliminating the need for such symbols and providing a straightforward and concise method for representing large numbers. The aim was to develop a more efficient method, arrow-free notation, reducing the complexity and steps necessary with previous notations.