Topics in Number Theory

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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.