  Euclidean distance matrices corresponding to an arithmetic progression have
rich spectral and structural properties. We exploit those properties to develop
completely positive factorizations of translations of those matrices. We show
that the minimal translation that makes such a matrix positive semidefinite
results in a completely positive matrix. We also discuss completely positive
factorizations of such matrices over the integers. Methods developed in the
paper can be used to find completely positive factorizations of other matrices
with similar properties.
