If it were possible to find solutions such as the ones above for enough different moduli, forming a complete covering system of congruences, the problem would be solved. However, as showed, a polynomial identity that provides a solution for values of congruent to mod can exist only when is not a quadratic residue modulo . For instance, is a not a quadratic residue modulo , so the existence of an identity for values of that are congruent to modulo does not contradict Mordell's result, but is a quadratic residue modulo so the result proves that there can be no similar identity for "all" values of that are congruent to modulo . As is a quadratic residue modulo for all , there can be no complete covering system of modular identities for all .