Thanks to John McKay's suggestion, I e-wrote Buium and this is what he replied: << I don't know if the project of the 2004 Talent search is related to the derivatives of numbers that I'm using. What I do is to take the derivative of a number x with respect to a prime number p as being the number "dx/dp":=(x-x^p)/p which is an integer due to Fermat's Little theorem. The fact that this is a good definition can be seen from a number of papers I wrote about it. The first of them appeared in Inventiones Math., 1995. Other relevant papers appeared in Duke Math J., 1996, Crelle J., 2000, etc. The precise references to the papers are listed on my web page www.math.unm.edu/~buium
From the wording of the PR squibs about this student's project, it seems likely to me that the number derivative she worked on is in fact Buium's. If any of you number theoretic folks care to take a look at Buium's papers to see just what the purpose of this definition is, and why it's called a "derivative", I'd like to hear what you have to say.
--Dan