Number Theory With Emphasis on Algorithms and Algebraic Number Theory

数论，重点是算法和代数数论

• 批准号: 9732709
• 负责人: Hendrik Lenstra
• 金额: $ 24.32万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9732709&HistoricalAwards=false
• 关键词: Number; Theory; Emphasis; Algorithms; Algebraic

ABSTRACT Hendrik Lenstra Berkeley 9732709 Professor Lenstra will study a collection of problems in algorithmetic number theory, algebraic number theory, and algebraic geometry. Topics to be studied include smooth numbers, Mersenne primes, Galois groups of local fields, Belyi functions, Lacunary polynomials, and polynomials over finite fields. Number theory is one of the oldest branches of mathematics. It is the study of the natural numbers, the basic numbers of counting. From the very beginning of the area, there has been a gap between what was known to be possible for all numbers, and what could actually be done for a specific one. For example, it is known that any natural number can be factored in only one way as the product of smaller unfactorable numbers. Yet given a carefully chosen large number, it can be so difficult to factor into parts that scientists have developed secure codes based on factorization. This project is devoted to improving the methods we have for actually computing some of the most important mathematical properties of particular numbers. In the computer age, the potential applications for improved methods of computation go well beyond mathematics and science and into everyday life.

摘要 Hendrik Lenstra Berkeley 9732709 Lenstra 教授将研究算法数论、代数数论和代数几何中的一系列问题。要研究的主题包括光滑数、梅森素数、局部域的伽罗瓦群、Belyi 函数、缺位多项式和有限域上的多项式。 数论是数学最古老的分支之一。 它是对自然数和计数的基本数字的研究。 从该领域一开始，已知的所有数字的可能性与针对特定数字实际可以做的事情之间就存在差距。 例如，众所周知，任何自然数只能以一种方式因式分解为较小的不可分解数的乘积。 然而，考虑到精心选择的大量数字，将其分解为多个部分可能非常困难，因此科学家们开发了基于分解的安全代码。 该项目致力于改进我们实际计算特定数字的一些最重要的数学属性的方法。 在计算机时代，改进计算方法的潜在应用远远超出了数学和科学的范围，进入了日常生活。