Topics in Number Theory

Pommerance 9701101该奖项支持在分析、计算和加法数论方面的工作。其中一位主要调查人员将进一步探索一些已知算法背后的基本思想，方法是对一些已建立的算法进行严格的研究，并开发必要的适当工具来帮助分析下一代算法。他还将研究素数域上有界度的有限域的乘法群的严格离散对数算法，以及解析数论中的几个‘鄂尔多斯类型’问题。另一位首席研究员将试图通过证明最大特征和的分布结果和下界来获得对特征和的更精确的理解。他还将研究ABC猜想的后果。最后，他将研究几个与椭圆曲线有关的臭名昭著的问题。这是数论领域的研究。数论始于整数和诸如一个整数被另一个整数整除的问题。它是最古老的数学领域之一，最初是出于纯粹的美学原因。然而，在过去的半个世纪里，它已经成为开发计算机科学的新算法和电子产品的新纠错码的必要工具。

Pommerance 9701101 This award supports work on problems in analytic, computational, and additive number theory. One of the principal investigators will probe further the basic ideas behind some known algorithms, by making rigorous some of the established algorithms, and by developing the appropriate tools necessary to help analyze the next generation of algorithms. He will also study rigorous discrete logarithm algorithms for the multiplicative group of a finite field with bounded degree over its prime field, as well as several 'Erdos-type" problems in analytic number theory. The other principal investigator will try to obtain a more precise understanding of character sums by proving distribution results and lower bounds for the largest such sums. He also will study consequences of the abc-conjecture. Finally he will examine several of the infamous problems involving elliptic curves. This is research in the field of number theory. Number theory starts with the whole numbers and questions such as the divisibility of one whole number by another. It is among the oldest fields of mathematics and it was originally pursued for purely aesthetic reasons. However, within the last half century, it has become an essential tool in developing new algorithms for computer science and new error correcting codes for electronics.