Topics in combinatorial and algorithmic number theory
组合和算法数论主题
基本信息
- 批准号:0703850
- 负责人:
- 金额:--
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:2007
- 资助国家:美国
- 起止时间:2007-07-01 至 2010-12-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Abstract for the award DMS-0703850of PomeranceThis proposal deals with several problems in computational and combinatorial number theory, and the theory of arithmetic functions with applications to ranks of elliptic curves. In the first category, the PI will work with Hendrik Lenstra on improving the AKS primality test, with applications to the construction of finite fields, and he will work with Lenstra and Jonathan Pila on a rigorously fast factorization algorithm involving jacobian varieties of hyperelliptic curves of genus 2. In combinatorial number theory, the PI will continue his recent work with Michael Filaseta, Kevin Ford, Sergei Konyagin, and Gang Yu dealing with the covering congruences problem of Paul Erdos.Among the arithmetic functions that the PI will study are the order of the multiplicative group modulo an integer (Euler's function) and the order of the largest cyclic subgroup of this group (Carmichael's function), including the index of this subgroup in the full group. This index has applications to the work of Ulmer on the ranks of elliptic curves over function fields of positive characteristic and some questions of Arnold. The PI will be working jointly with Igor Shparlinski on the former applications and with Michel Balazard on the latter. With the Carmichael function, the PI intends with Florian Luca to strongly improve the known results concerning its range.One would think that basic questions concerning the integers, such as deciding when one is prime, factoring another, or the multiplicative structure of the remainders when a particular number is used as a divisor, would be well-understood by now. But they are not, and for one of these problems, namely factoring, the fact that it is apparently difficult in general is the basis for the security of cryptographic systems that underpin commerce and communication. The principal proposed activities of this proposal concern these fundamental problems, mostly from the theoretical side. In addition, the PI proposes to study some applications of the multiplicative structure of remainders to the study of the ranks of certain elliptic curves. Elliptic curves arise from cubic equations in two variables; one studies the solutions to such an equation with the variables coming from a given algebraic domain such as the rational numbers and similar domains. They have provided a rich source of structure for number theorists. These curves too have been used in cryptographic systems and in computational number theory, but the PI's focus here is a statistical study of certain algebraic invariants of the solution sets as one varies curve parameters. Finally, the PI will study some problems in combinatorial number theory, including the famous covering congruences problem of the late Paul Erdos. Here one asks if there can be a finite set of large integers and a single remainder for each such that every integer when divided by the given large numbers leaves at least one corresponding remainder in the given pre-set list. The current record for "large" is 25, and it is not known if one can do this with arbitrarily large numbers.
摘要奖DMS-0703850的PomeranceThis建议涉及计算和组合数论中的几个问题,以及算术函数的理论与应用程序的行列椭圆曲线。 在第一类中,PI将与Hendrik Lenstra合作改进AKS素性测试,并将其应用于有限域的构造,他将与Lenstra和Jonathan Pila合作研究一种严格快速的因子分解算法,涉及亏格2的超椭圆曲线的雅可比变种。 在组合数论方面,PI将继续他最近与Michael Filaseta,Kevin福特,Sergei Konyagin和Gang Yu一起处理Paul Erdos的覆盖同余问题的工作。PI将研究的算术函数中包括乘法群模整数的阶(欧拉函数)和这个群的最大循环子群的阶(卡迈克尔函数),包括这个子群在全群中的指数。 这个指数的应用工作的乌尔默的行列椭圆曲线的功能领域的积极特征和一些问题阿诺德。 PI将与Igor Shparlinski合作开发前一种应用程序,并与Michel Balazard合作开发后一种应用程序。 利用卡迈克尔函数,PI打算与Florian Luca一起大力改进关于其范围的已知结果。人们可能会认为,关于整数的基本问题,例如决定一个整数何时是素数,因式分解另一个整数,或者当一个特定的数被用作除数时,整数的乘法结构,现在应该已经很好地理解了。 但事实并非如此,对于其中一个问题,即保理,它显然是困难的,这一事实是支撑商业和通信的密码系统安全的基础。 本建议的主要活动涉及这些基本问题,主要是理论方面的问题。 此外,PI建议研究乘法结构的一些应用,以研究某些椭圆曲线的秩。 椭圆曲线产生于两个变量的三次方程;一个研究这样一个方程的解,变量来自一个给定的代数域,如有理数和类似的域。 它们为数论提供了丰富的结构来源。这些曲线也被用于密码系统和计算数论中,但PI在这里的重点是随着曲线参数的变化,对解集的某些代数不变量进行统计研究。最后,PI将研究组合数论中的一些问题,包括已故Paul Erdos的著名覆盖同余问题。这里我们要问的是,是否可以有一个大整数的有限集合,每个整数都有一个余数,使得每个整数除以给定的大数时,在给定的预设列表中至少留下一个对应的余数。 目前“大”的记录是25,不知道是否可以用任意大的数字来做。
项目成果
期刊论文数量(0)
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Carl Pomerance其他文献
An inequality related to the sieve of Eratosthenes
- DOI:
10.1016/j.jnt.2023.07.005 - 发表时间:
2024-01-01 - 期刊:
- 影响因子:
- 作者:
Kai Fan;Carl Pomerance - 通讯作者:
Carl Pomerance
On a nonintegrality conjecture
- DOI:
10.1007/s40879-021-00507-3 - 发表时间:
2021-10-12 - 期刊:
- 影响因子:0.500
- 作者:
Florian Luca;Carl Pomerance - 通讯作者:
Carl Pomerance
On the Distribution in Residue Classes of Integers with a Fixed Sum of Digits
- DOI:
10.1007/s11139-005-0824-6 - 发表时间:
2005-03-01 - 期刊:
- 影响因子:0.700
- 作者:
Christian Mauduit;Carl Pomerance;András Sárközy - 通讯作者:
András Sárközy
On Locally Repeated Values of Certain Arithmetic Functions, IV
- DOI:
10.1023/a:1009723712317 - 发表时间:
1997-01-01 - 期刊:
- 影响因子:0.700
- 作者:
Paul Erdös;Carl Pomerance;András Sárközy - 通讯作者:
András Sárközy
On primes and practical numbers
- DOI:
10.1007/s11139-020-00354-y - 发表时间:
2021-02-15 - 期刊:
- 影响因子:0.700
- 作者:
Carl Pomerance;Andreas Weingartner - 通讯作者:
Andreas Weingartner
Carl Pomerance的其他文献
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{{ truncateString('Carl Pomerance', 18)}}的其他基金
Problems in Algorithmic and Combinatorial Number Theory
算法和组合数论中的问题
- 批准号:
0401422 - 财政年份:2004
- 资助金额:
-- - 项目类别:
Continuing Grant
Mathematical Sciences: Topics in Analytic and Algorithmic Number Theory
数学科学:解析和算法数论主题
- 批准号:
9206784 - 财政年份:1992
- 资助金额:
-- - 项目类别:
Continuing Grant
Mathematical Sciences: Topics in Analytic and Algorithmic Number Theory
数学科学:解析和算法数论主题
- 批准号:
9002538 - 财政年份:1990
- 资助金额:
-- - 项目类别:
Continuing Grant
Mathematical Sciences: Topics in Number Theory
数学科学:数论主题
- 批准号:
8803297 - 财政年份:1988
- 资助金额:
-- - 项目类别:
Continuing Grant
High Speed Factoring with the Quadratic Sieve Algorithm and a Pipeline Architecture
使用二次筛算法和管道架构进行高速分解
- 批准号:
8702941 - 财政年份:1987
- 资助金额:
-- - 项目类别:
Standard Grant
High Speed Factoring with the Quadratic Sieve Algorithm and a Pipeline Architecture (Mathematical Sciences and Computer Research)
使用二次筛算法和管道架构进行高速因式分解(数学科学和计算机研究)
- 批准号:
8421341 - 财政年份:1985
- 资助金额:
-- - 项目类别:
Continuing Grant
Mathematical and Computer Sciences: Computational and Multiplicative Number Theory
数学和计算机科学:计算和乘法数论
- 批准号:
8301487 - 财政年份:1983
- 资助金额:
-- - 项目类别:
Standard Grant
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基于诱导ES细胞定向分化的化合物库构建和信号转导分子事件发现
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