"Balanced Sets" - 

The decomposition of the positive integers up to 8 into the sets
{1,4,6,7} and {2,3,5,8} is unusually well balanced in the
sense that the k-th moments of the two sets are equal for
k = 0, 1, 2, where the k-th moment of a set is defined, here, to be
the sum of the k-th powers of its elements.  The question of when
the integers up to n can be divided into two m-balanced sets
(i.e., their first m moments agree) is motivated by applications
in signal processing and error-correcting codes (and, as will be
explained, by several much more fanciful applications).  As will be
explained, this is equivalent to finding for which n there is a
polynomial in x, all of whose coefficients are +/-1, that is
divisible by (x-1)^m.  New results on this problem will be described.

This is joint work with Shahar Golan, Rob Pratt, and Stan Wagon.