  We introduce an equation defined on a multi-dimensional lattice, which can be
considered as an extension to the coprimeness-preserving discrete KdV like
equation in our previous paper. The equation is also interpreted as a
higher-dimensional analogue of the Hietarinta-Viallet equation, which is famous
for its singularity confining property while having an exponential degree
growth. As the main theorem we prove the Laurent and the irreducibility
properties of the equation in its "tau-function" form. From the theorem the
coprimeness of the equation follows.
  In Appendix we review the coprimeness-preserving discrete KdV like equation
whichis a base equation for our main system and prove the properties such as
the coprimeness.
