Self-conjugate t-core partitions, sums of squares, and p-blocks of An

We prove that if t is an integer with t=8 or t 10, then every integer n>2 has a self-conjugate t-core partition. This result has consequences in the representation theory of alternating groups, and has a version as a theorem about the representation of integers by sums of squares. We also give an infinite sequence of integers that have no self-conjugate 9-core partition.