计算机辅助的拉马努贾-萨托数列的构造为1分π
Ralf Hemmecke1, Peter Paule1,2, Cristian-Silviu Radu1
1Research Institute for Symbolic Computation (RISC), Johannes Kepler University, 4040 Linz, Austria.
概括
本研究介绍了"萨托构造",一种算法方法,用于生成1/π的拉马努贾-萨托数列. 这些系列,包括著名的例子,是严格证明使用先进的算法,如MultiSamba.
科学领域:
- 数学理论 数学理论
- 计算数学 计算数学 计算数学
- 代数几何几何学的几何学
背景情况:
- 萨托和 (2004) 介绍了一种使用模块化形式构建1/π数列的方法.
- 现有的数列,比如拉马努贾和朱德诺夫斯基兄弟的数列,对于计算π至关重要.
- 对于这些序列,人们认识到对算法和经过严格验证的方法的需求.
研究的目的:
- 介绍萨托和的构造的算法版本,称为"萨托构造".
- 介绍和分析由这种算法方法生成的"拉马努贾 - 萨托"序列.
- 用计算工具证明这些数列的严格,非数值证明.
主要方法:
- 基于模块化形式的"Sato构造"算法的开发.
- 使用无限家族的萨托三倍数 (N, γN, τN) 进行序列生成.
- 应用"MultiSamba"算法 (Samba的扩展) 来评估在虚象二次点上的模块函数.
- 与全方位工具箱程序和用于证明序列身份的"ModFormDE"算法的集成.
主要成果:
- 成功实现了算法"萨托构造",用于生成拉马努贾-萨托序列.
- 证明 1/π 的著名数列符合这个框架.
- 使用MultiSamba算法生成的所有数列的严格,非数值证明.
结论:
- 萨托构造提供了一种系统和算法方法,用于生成1/π的序列.
- 多Samba算法是一个强大的工具,用于证明模块化函数的评估,从而证明了1/π的序列.
- 这项工作提供了一个严格的框架,通过模块化形式和计算算法来发现和验证数学常数.
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