  This paper has two main parts. In the first part we develop an elementary
coordinatization for any nilpotent group $G$ taking exponents in a binomial
principal ideal domain (PID) $A$. In case that the additive group $A^+$ of $A$
is finitely generated we prove using a classical result of Julia Robinson that
one can obtain a central series for $G$ where the action of the ring of
integers $\Z$ on the quotients of each of the consecutive terms of the series
except for one very specific gap, called the special gap, is interpretable in
$G$. Then we use a refinement of this central series to give a criterion for
elementary equivalence of finitely generated nilpotent groups in terms of the
relationship between group extensions and the second cohomology group.
