Title: p-adic Numbers and Their Applications: A whole new way to think of convergence

Speaker: David Manderscheid (UTK)

Abstract: 

A sequence of rational numbers that converges may not converge to a rational number. For example, the sequence 3, 3.1, 3.14, 3141, 3.1415…. and so on, where each term is obtained by taking the next term in the decimal expansion of π, converges to π. More generally, every real number can be thought of as the limit of a sequence of rational numbers via its decimal expansion. Thus, it is much more “rational” to do calculus on the set of real numbers than the rational numbers. Moreover, the set of real numbers is the “smallest” set where calculus can be done.

But what if we are interested in doing number theory, the branch of mathematics that studies the properties of rational numbers and integers and includes concept such as prime numbers, factorization (essential to cryptography), and integer solutions to polynomial equations (e.g, Fermat’s Last Theorem)? Might there be other ways to think of convergence that reflect number theory more than they do calculus? The answer is a resounding yes, as I will explain. To this end, I will introduce the p-adic numbers, where p is a prime number, and give some applications. No knowledge of number theory will be assumed – yet we will get to some amazing theorems.