Let $Q$ be a quadratic form in $n$ variables with integer coefficients.  Let us say that $Q$ has the "special property" mod $p$, if the relation $Q(x_1,...,x_n)=0$ (mod $p$) implies that $(x_1,...,x_n)=(0,...,0)$ (mod $p$).  (There must be a name for this property, but I don't know it, which is why I'm calling it "the special property".)  Let us say that $Q$ is "special infinitely often" if there are infinitely many primes $p$ such that $Q$ has the special property mod $p$.  For example, the one-variable quadratic form $Q(x)=x^2$ is special infinitely often.  Another simple example is that $Q(x,y) = x^2 + y^2$ is special infinitely often because it is special mod $p$ whenever -1 is a quadratic non-residue mod $p$.  In fact, any two-variable quadratic form that is non-degenerate over the rationals is special infinitely often because $Q(x,y) = ax^2 + bxy + cy^2$ is special whenever $p$ is odd and the discriminant $b^2 - 4ac$ is a quadratic non-residue mod $p$.

I've been trying to find a quadratic form in more than two variables that is special infinitely often, but I'm doubtful that such a thing exists.  As far as I know, each of the following statements could either be valid or invalid.  (Although obviously [c] implies [b] implies [a].)

[a]  For every integer quadratic form $Q$ in three or more variables, the set of $p$ such that $Q$ has the special property mod $p$ is finite.

[b]  The set of primes $p$ such that there exists an integer quadratic form $Q$ in three or more variables having the special property mod $p$ is finite.

[c]  The set of primes $p$ such that there exists an integer quadratic form $Q$ in three or more variables having the special property mod $p$ is empty.

Can anyone help resolve any of these questions?