The period length of Euler's number e

TL;DR

This paper studies the periodicity of Jacobi symbol sequences derived from continued fraction convergents of Euler's number e and its square, revealing minimal period lengths and initiating a general theory.

Contribution

It identifies the minimal period lengths for the Jacobi symbol sequences of e and e^2 and explores the diversity of possible periods across real numbers.

Findings

Shortest period length for e is 24.

Shortest period length for e^2 is 40.

Uncountably many numbers have period 1.

Abstract

Let s_k/t_k, k>= 0, be the convergents of the continued fraction expansion of a real number x. We investigate the sequence of Jacobi symbols (s_k/t_k), k>= 0. We show that this sequence is purely periodic with shortest possible period length 24 for x=e=2.718281... and shortest possible period length 40 for x=e^2. Further, we make the first steps towards a general theory of such sequences of Jacobi symbols. For instance, we show that there are uncountably many numbers x such that this sequence has the period 1 (of length 1), and that every natural number L actually occurs as the shortest possible period length of some x.