关于所有数字为偶数的二的幂的有限性研究
数论
2025-08-13 v3 动力系统
摘要
我们研究寻找正整数 n 的问题，使得 2^n 的所有十进制数字都是偶数，即属于 {0, 2, 4, 6, 8}。计算检查至 n = 10^15 后发现已知案例 n = 1, 2, 3, 6, 11 且无额外实例。我们提出了自包含的论证，基于动力学 Borel-Cantelli 引理，建立了与此问题相关的度量结果。我们证明生成无限多个此类幂的初始相位集合在李普希茨测度下为零，为有限性猜想提供了强大的概率论支持。
引用
@article{arxiv.2503.23177,
title = {Notes on the Finiteness of Powers of Two with All Even Digits},
author = {Bogdan C. Dumitru},
journal= {arXiv preprint arXiv:2503.23177},
year = {2025}
}
备注
This submission replaces the withdrawn v1. The original version contained a critical error in its main proof. This new version corrects the error, replacing the original unsubstantiated claim with a rigorous metric result and clarifying that the specific problem for powers of two is an open conjecture