http://www.google.com/search?hl=en&ie=ISO-8859-1&q=exp+pi+sqrt+integer has some pages with explanations involving the Jacobi function J, elliptic curves and other mathematical esoterica. Unfortunately it does not seem likely that an explanation suitable for a college senior is available, unless he is willing to educate himself for several months on higher mathematics.

Hal Finney > Exp(Pi*Sqrt(n)) Page > This table lists values of Exp(Pi*Sqrt(n)), for some selected values of > n up to 1000. Some of these values are very close to integers. A prize > will be awarded to anyone who can either convincingly argue that this is > coincidence, or who can explain why this is so in terms intelligible to > an intelligent college senior. Something else that might help to lift > the veil on this mystery would be a predictor: for a given value of n, > is Exp(Pi*Sqrt(n)) close to an integer? > ... > 163 262537412640768743.9999999999992 > ...