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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?

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1 Answer 1

up vote 10 down vote accepted

The Chevalley–Warning theorem (see wikipedia) implies that any quadratic form in at least three variables has a non-trivial solution modulo p.

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Thanks! That settles it. –  Bobby Kleinberg Mar 7 '10 at 22:49

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