The period length of Euler's number e
Number Theory
2013-03-13 v1
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.
Cite
@article{arxiv.1303.2887,
title = {The period length of Euler's number e},
author = {Kurt Girstmair},
journal= {arXiv preprint arXiv:1303.2887},
year = {2013}
}
Comments
11 pages