A geometric proof that $e$ is irrational and a new measure of its irrationality

TL;DR

This paper presents a simple geometric proof of the irrationality of e and introduces a new measure of its irrationality, relating it to prime factors and conjectures.

Contribution

It provides a novel geometric proof of e's irrationality and proposes a new quantitative measure of its irrationality based on factorial divisibility.

Findings

e is irrational, proven via nested intervals

New lower bound for |e - p/q| involving factorials and prime factors

Connections made between the measure of irrationality and prime factors

Abstract

We give a simple geometric proof that $e$ is irrational, using a construction of a nested sequence of closed intervals with intersection $e$. The proof leads to a new measure of irrationality for $e$: if $p$ and $q$ are integers with $q > 1$, then $|e - p/q| > 1/(S(q)+1)!$, where $S(q)$ is the smallest positive integer such that $S(q)!$ is a multiple of $q$. We relate this measure for $e$ to a known one and to the greatest prime factor of an integer. We make two conjectures and recall a theorem of Cantor that can be proved by a similar construction.