  Based on the MRDP theorem concerning the Hilbert tenth problem, there is a
corresponding Diophantine equation called proof equation for every formula of
the First-order Peano Arithmetic (PA). A formula is provable in PA, if and only
if the corresponding proof equation has solution. Based on proof equation, some
famous sentences, e.g., the Godel sentence, the Rosser sentence and the Henkin
sentence, can be expressed by the form of Diophantine equation. It is proved
that for every axiom and theorem in the PA, we can construct actually a
corresponding Diophantine equation, for which we know that it has no solution,
but this fact cannot be proved in the PA. This means that, for every axiom and
theorem in the PA, we can construct actually a corresponding undecidable
proposition. Finally, generalizing the idea of proof equation to any
mathematical (set theoretical, number theoretical, algebraic, geometrical,
topological, et al) proposition, a project translating the task seeking a proof
of the mathematical proposition into solving a corresponding Diophantine
equation is discussed.
