Graduate Math Seminar

Title: What Happens If We Keep Squaring 5?

Abstract: Consider the sequence obtained by repeatedly squaring 5:

5, 25, 625, 390625, 152587890625, ...

The numbers quickly become enormous, yet something curious begins to happen: their digits start settling down from the right, as though the sequence were approaching a hidden limit. 

But approaching in what sense?

A number is not intrinsically “large”, “small”, “near”, or “far”. These words acquire meaning only after we choose a way of measuring. We are all familiar with the usual absolute value, which measures the magnitude of a number. In this talk, we will discover the only other natural way to measure the rational numbers. We will explore the strange arithmetic and geometry they create, and see why they have become indispensable in modern number theory. 

Welcome to the world of p-adic numbers!