Being a graduate student in mathematics can be a wonderful experience. I suppose the same is probably true in any discipline. The life of a grad student is largely free of the responsibilities that most adults must bear, and so it is a kind of extended childhood. Moreover, although one must work very hard to learn the material well enough to become a mathematician, most students love the subject enough to want to put that effort in...and when that isn't enough, there is the comraderie of bearing that burden together. Finally, the students develop very different roles (the social one who keeps everyone entertained, the brilliant one who seems to have all the answers, the connected one who has contact with lots of real mathematicians, etc.) I tried to capture as much of that as I could in this short story.

I don't think I would have considered the idea of a "secret message" hidden in a number had it not been for Carl Sagan's novel Contact. In the book, but not the movie, Ellie is told by aliens that they found a secret message hidden in one of the constants of mathematics. Consider, for instance, if you found a string of digits in the expansion of e or π that looked like a coded message. What would that mean? Who could have left such a message?

To play with this idea, I use two real pieces of mathematics. On the one hand, there is a recently discovered algorithm for finding any given digit of the number π when expanded in hexadecimal (see here, for instance). And, on the other hand, there is an old result of Hardy and Littlewood which shows that any given finite sequence of digits is almost certain to appear in the decimal expansion of any randomly selected irrational number. (That is, although there are examples of real numbers which do not contain your phone number in their decimal expansion, if you select one at random then the probability that you will pick one that does not contain your phone number is vanishingly small.)

When you combine these, you get the surprising fact that the number π almost certainly contains any given sequence of digits (for instance, it probably contains the text of this very webpage translated into digits by using the ASCII code for each character). On the other hand, finding the location of that string of digits in π by using the Bailey-Borwein-Plouffe formula is practically impossible!

So, again, this story was an opportunity to play around with these ideas in the context of the social life of some math grad students. I'm curious to know whether other people enjoyed it, found it interesting, found it confusing, or what. Please write in with your comments if you've read it!