NAME:  Dennis Eichhorn
 
 ADDRESS:  Department of Mathematics
   University of Arizona
   Tucson, AZ 85721-0001

 EMAIL ADDRESS:  eichhorn@math.arizona.edu
 
 TITLE OF TALK:  On the divisibility of $r_k(n)$, the number of
                  representations of $n$ as a sum of $k$ squares.
 
 ABSTRACT OF TALK:  In some recent work by the speaker, a technique for
                     proving congruences for $r_k(n)$ in arithmetic 
                     progressions is developed.  In particular, an
                     explicit constant $C$ is produced such that if a 
                     congruence of the form $r_k(hn+r) \equiv 0 \pmod m$
                     holds for all $n \leq C$, then the congruence must
                     hold for all $n$.  This follows from some elementary 
                     properties of modular forms.  Bateman has now given
                     an elementary proof of a slightly stronger version of 
                     certain cases of this theorem, and in doing so, he 
                     has shown that in some instances, the constant $C$ 
                     produced using the former technique is best possible.  
                     In this talk, we discuss these two new insights into 
                     $r_k(n)$.