欧拉数e的周期长度
The period length of Euler's number e
摘要：
设s_k/t_k，k>= 0，是实数x的连分数展开式的收敛性。本文研究Jacobi符号序列（s_k/t_k），k>= 0.我们表明，这个序列是纯周期性的最短可能的周期长度为24 x=e=2.718281.以及x=e^2的最短可能周期长度40。此外，我们对这样的Jacobi符号序列的一般理论的第一步。例如，我们证明有不可数多个数x使得这个序列的周期为1（长度为1），并且每个自然数L实际上都是某个x的最短可能周期长度。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
