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

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.

arXiv comment7 pages, 1 figure, Addendum gives details on why the intersection of the intervals is $e$

References in corpus (1)

• Irrationality Measures, Irrationality Bases, and a Theorem of Jarnik