Diophantine equation
Martin Davis studied algorithmic questions involving the number of solutions of a Diophantine equation. Hilbert's tenth problem asks whether or not that number is 0. Let formula_45 and let formula_46 be a proper non-empty subset of formula_30. Davis proved that there is no algorithm to test a given Diophantine equation to determine whether the number of its solutions is a member of the set formula_46. Thus there is no algorithm to determine whether the number of solutions of a Diophantine equation is finite, odd, a perfect square, a prime, etc.