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$x^2 + y^2 = -1$ is a weird equation because it has no solutions over $\Bbb R$. I want to count the number of solutions it has over $\Bbb Z / p$ where $p$ is prime.

If $p = 2$ then it has $p$ solutions. This is to do with the fact that squaring is a field automorphism.

If $p \equiv 1 \pmod{4}$ then there is an $i$ such that $i^2 = -1$ so $$x^2 + y^2 = -1 \implies \left({x \over i}\right)^2 + \left({y \over i}\right)^2 = 1 \implies \left({x \over i} + y \right)\left({x \over i} - y \right) = 1$$ which has $p - 1$ solutions.

If $p \equiv 3 \pmod{4}$ then the situation is more complicated. The thing I noticed is that $A = \{x \mid x^2 + y^2 = -1\}$ and $B = \{x \mid y^2 - x^2 = 1\}$ form a partition of $\Bbb Z/p$. Reason being that $x \not\in A \implies (-1 - x^2 \mid p) = -1 \implies (1 + x^2 \mid p) = 1 \implies (\exists y)\,1+x^2 = y^2 \implies x \in B$, and vice versa. Also notice that $A \cap B = \emptyset$. So we get $|A| = |\Bbb Z / p| - |B| = p - |B|$. To determine $|B|$, use the fact that for every $(x,y)$ for which $y^2 - x^2 = 1$, $(x,-y)$ also satisfies the equation, so $|B|$ is the number of solutions to $y^2 - x^2 = 1$ divided by $2$, which is $\frac{p-1}{2} \therefore \,|A| = {p + 1 \over 2}$. Now it's easy to see that the number of solutions to $x^2 + y^2 = -1$ is $2|A|$ which is $p+1$.

Any quicker method?

wlad
  • 8,185

2 Answers2

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Given a finite field $F$, let $C = F[i]/(i^2+1)$. Define the mapping $N:C \to F$ by $N(x+iy) = x^2 + y^2$ and note that it's a multiplicative homomorphism from $C^*$ to $F^*$.

Now let $X = \{x \in C \mid N(x) = 1\}$ and $Y = \{x \in C \mid N(x) = -1\}$. The aim is to determine $|Y|$.

Note that there exists an $e \in Y$. I will omit the proof.

Define the mapping $f: X \to Y$ by $f(x) = ey$. First, the codomain is correct because given $x \in X$, $N(ex) = N(e)N(x) = (-1) \times 1 = -1$. Second, the mapping is injective because $f(x) = f(x') \implies ex = ex' \implies x = x'$. Finally, the mapping is surjective because given $y \in Y$, $N\left({y \over e}\right) = 1$ so ${y \over e} \in X$ and $f\left(y \over e\right) = e{y \over e} = y$.

So $f$ is a bijection. This implies that we only need to count $X$ to find $|Y|$. Here are some ways to do that.

Community
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wlad
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We want to count the number of points on a curve $x^2+y^2=-1$. Let's first count the number of points on the corresponding curve in the projective space and then subtract the number of points at infinity.

The first part is solved by «rational parametrization»: since our curve $X^2+Y^2=-Z^2$ has degree 2, any line that intersects it intersects it at exactly two points; so for a point $O$ on the curve any line through $O$ intersects the curve at exactly one more point — i.e. a conic has $p+1$ points (and is isomorphic to $\mathbb P^1(\mathbb F_p)$, whatever it means).

Now the rest is easy: at infinity (i.e. on the line $Z=0$) we have equation $X^2+Y^2=0$ — so there are 2 points if $(-1/p)=1$ and no points otherwise.

This gives the answer $p-(-1/p)$.

P.S. Note that the same method works for any (non-degenerate) conic — e.g. for $x^2+y^2=1$ from your previous question.

Grigory M
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