  In this paper we first review the history of Hilbert's Tenth Problem, and
then study mixed quantifier prefixes over Diophantine equations with integer
variables. For example, we prove that $\forall^2\exists^4$ over $\mathbb Z$ is
undecidable, that is, there is no algorithm to determine for any
$P(x_1,\ldots,x_6)\in\mathbb Z[x_1,\ldots,x_6]$ whether $$\forall x_1\forall
x_2\exists x_3\exists x_4\exists x_5\exists x_6(P(x_1,\ldots,x_6)=0),$$ where
$x_1,\ldots,x_6$ are integer variables. We also have some similar undecidable
results with universal quantifies bounded, for example,
$\exists^2\forall^2\exists^2$ over $\mathbb Z$ with $\forall$ bounded is
undecidable. We conjecture that $\forall^2\exists^2$ over $\mathbb Z$ is
undecidable.
