  We reformulate the singularity confinement of the discrete Toda equation. We
prove the co-primeness property, which has been introduced in our previous
paper (arXiv:1311.0060) as one of the integrability criteria, for the discrete
Toda equation. We study three types of boundary conditions (semi-infinite,
molecule, periodic) for the discrete Toda equation, and prove that the same
co-primeness property holds for all the types of boundaries. (v2: typos
corrected, final version to appear in J. Math. Phys.)
