I have had trouble understanding class numbers. Usually they measure the "failure" of unique factorization. So how does these failures become a special type of number, that multiply to form a group?

Here is an article explaining why exp(pi*sqrt(163)) = is near an integer: http://mathoverflow.net/questions/4775/why-are-powers-of-exp... http://math.stackexchange.com/questions/4544/why-is-e-pi-sqr...

I have met all three of the mathematicians pictured.