Nearly 40 years ago, Bousfield and Ravenel noticed that the operations on p-adic complex K-theory are abstractly closely related to the Iwasawa algebra.  In the mid-90's, Dwyer and Mitchell used this observation to describe the topological K-theory of the algebraic K-theory of the integers in terms of p-adic L-functions.  In this talk, I will give a gentle introduction to this amazing story and then explain work with Mike Mandell which extends these results and suggests a program for using topological information to answer number-theoretic questions.