We have performed density-functional theory calculations, including the spin-orbit correction, to investigate atomic gold chains on Ge(001). A set of 26 possible configurations of the Au/Ge(001) system with c(4×2) and c(8×2) symmetries is studied. Our data show that the c(4×2) order results in the lowest energy, which is not in direct agreement with recent experiments. Using total-energy calculations, we are able to explain these differences. We address the electronic band structure and apply the Tersoff-Hamann approach to correlate our data to scanning-tunneling microscopy (STM). We obtain two highly competitive structures of the atomic Au chains for which we report simulated STM images in order to clarify the composition of the experimental Au/Ge(001) surface.