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 Brian Conrey Director, American Institute of Mathematics (650) 845-2071 conrey@aimath.org

# San Jose State Math Researcher Experiences Epiphany at American Institute Of Mathematics

Dan Goldston to expound on his prime number theory result at Palo Alto math workshop

PALO ALTO, CA-March 21, 2003- Dan Goldston was standing in front of a whiteboard at the American Institute of Mathematics (AIM) last fall when a comment made by fellow mathematician Roger Heath-Brown sparked a brainwave. This inspiration enabled him to devise a completely new approach to a problem he had been researching for decades. Next week, Goldston will present the culmination of twenty years of research at the AIM-hosted workshop on Algorithmic Number Theory.

His paper, titled "Small Gaps Between Primes," and co-authored with Turkish mathematician Cem Yildirim, places mathematicians closer to the tantalizing goal of identifying the frequency and location of 'twin primes' - prime numbers that differ by two. Prime number research has long been the focus of gifted mathematicians. As early as the third century, B.C., the Greek mathematician Eratosthenes developed a way to systematically find the prime numbers. Since then, notable mathematicians such as Fermat (17th century), Riemann (1859), Hardy and Littlewood (1920s), and Bombieri and Davenport (1965) have contributed foundational theory on the pattern of prime numbers - numbers that cannot be divided by any number smaller than themselves (other than 1) without leaving a remainder.

Small primes are relatively easy to determine; it's the large prime numbers with which mathematicians have been wrestling. The smallest prime numbers are 2, 3, 5, 7, 11, 13, 17 and 19. Since prime numbers are the building blocks of the integers (they can be multiplied to obtain all of the other integers), these small primes are familiar to elementary school students. Anyone with an interest in patterns may observe that primes occur in twins with a surprising regularity. For example: 11,13; 17, 19; 29, 31; 41, 43; 59, 61.

Just as with primes, the frequency of twin primes decreases as one progresses to higher numbers. But do they completely fizzle out beyond some very large number? No one knows the answer for certain, but Goldston's new theory significantly advances mathematicians' knowledge of how primes are distributed, and even shines some light on the hard-to-identify location of very large prime numbers.

Goldston's presentation on Friday, March 28, will come at a timely moment. Some of the world's leading mathematicians will be in Palo Alto for a brainstorming session on Algorithmic Number Theory. The conference, one of a series to be hosted by AIM over the next few years, was assembled to examine and possibly extend the recent breakthrough in primality testing announced last year by computer scientist Manindra Agrawal of the Indian Institute of Technology in Kanpur, and his students Neeraj Kayal and Nitin Saxena.

Organizers are hopeful that AIM's whiteboard will once again be the catalyst for further breakthroughs - some of which could have enormous relevance to the mathematics that powers secure internet transactions.

"This new work of Goldston and Yildirim is a breakthrough that mathematicians interested in prime numbers have been looking for over the last 80 years. There will be a number of important developments that will follow from this innovation," said Brian Conrey, Executive Director of AIM.

A technical description of Goldston's new work can be found at: http://aimath.org/goldston_tech.

The need for a collaborative approach to mathematics
AIM is fast emerging as a world center for collaborative mathematics. Today, mathematics is flourishing because of a growing need to model complex phenomena and increased public awareness of its importance as the foundation for our technology-driven world. Thousands of mathematicians at hundreds of universities around the world are making seminal contributions to the subject. Whereas fifty years ago mathematical collaboration was relatively rare, today approximately half of all mathematical papers are written by multiple authors, with rich results for the field. Mathematics workshops, traditionally important in the development of mathematics, are emerging as crucial to continuing and extending this increased level of collaboration. AIM recently received a significant grant from the National Science Foundation for the express purpose of regularly convening focused mathematics workshops.

About the American Institute of Mathematics
The American Institute of Mathematics is a non-profit organization devoted to expanding the frontiers of mathematical knowledge and their applications, through focused research projects, sponsored workshops, and the development of a wide range of research tools made freely available over the internet. AIM is also involved in educational and outreach activities, especially those that further the integration of young people into mathematical research. AIM is headquartered in Palo Alto, California. For more information, visit http://www.aimath.org.