  Ehrhart polynomials are extensively-studied structures that interpolate the
discrete volume of the dilations of integral $n$-polytopes. The coefficients of
Ehrhart polynomials, however, are still not fully understood, and it is not
known when two polytopes have equivalent Ehrhart polynomials. In this paper, we
establish a relationship between Ehrhart-equivalence and other forms of
equivalence: the $\operatorname{GL}_n(\mathbb{Z})$-equidecomposability and
unimodular equivalence of two integral $n$-polytopes in $\mathbb{R}^n$. We
conjecture that any two Ehrhart-equivalent integral $n$-polytopes
$P,Q\subset\mathbb{R}^n$ are $\operatorname{GL}_n(\mathbb{Z})$-equidecomposable
into $\frac{1}{(n-1)!}$-th unimodular simplices, thereby generalizing the known
cases of $n=1, 2, 3$. We also create an algorithm to check for unimodular
equivalence of any two integral $n$-simplices in $\mathbb{R}^n$. We then find
and prove a new one-to-one correspondence between unimodular equivalence of
integral $2$-simplices and the unimodular equivalence of their $n$-dimensional
pyramids. Finally, we prove the existence of integral $n$-simplices in
$\mathbb{R}^n$ that are not unimodularly equivalent for all $n \ge 2$.
