Primes Between Squares -- Commentary on Appendix 8 of Laws Of Form

TL;DR

This paper analyzes Spencer-Brown's proof of the conjecture that at least two primes exist between consecutive squares, blending standard arguments with original insights into number behavior.

Contribution

It provides a detailed commentary and guide to Spencer-Brown's unique proof of a longstanding number theory conjecture, highlighting its originality and implications.

Findings

Spencer-Brown's proof combines rigorous and conjectural arguments.

The paper offers insights into the nature of numbers and primes.

It clarifies a complex proof for broader understanding.

Abstract

This paper provides a commentary and guide to Appendix 8 of Laws Of Form, which is a chapter (appendix) on number theory in the book Laws of Form by Spencer-Brown. (Spencer-Brown,Laws Of Form,Revised Seventh English edition. Bohmeier Verlag. 2020) This chapter in the book provides Spencer-Brown's proofs of the conjecture that there are at least two prime numbers between any consecutive squared numbers. That there are primes between squares has been a conjecture in number theory since Legendre. In Spencer-Brown's appendix he gives his proofs of the conjecture. Those proofs are a highly original mixture of standard rigorous arguments and also some stated facts about the way numbers behave that would be considered conjectures by most number theorists. These phenomena are very interesting and constitute a deep observation about the nature of number itself. We intend that our guide will enable the reader to gain insight into Spencer-Brown's point of view and that our discussions will be of interest to anyone with curiosity about the theory of numbers.