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Abstract
Prime numbers are building blocks in the theory of numbers, as well as in many of its applications to the digital world. Even so, many of the simplest long-standing questions about primes remain unsolved.
We review some of these questions and the tools that have been developed to address them, particularly a robust ‘Sieve method’ that applies to geometric configurations. Besides the modern mathematical tools that are involved in this sieve, notions from information science also play a decisive role. In this lecture, Professor Sarnak will illustrate the theory with concrete applications to integral Pythagorean triangles, integral Apollonian packings, and Markoff triples.
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