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Let $p$ be a prime. How many solutions $(x, y)$ are there to the equation $x^2 + 3y^2 \equiv 1 \pmod{p}$? Here $x, y \in \{0, 1, \ldots p-1\}$. This paper (https://arxiv.org/abs/1404.4214) seems like it may give some ideas, but focuses on equations of the form $\sum_{i = 1}^n x_i^2 \equiv 1 \pmod{p}$, i.e. it does not allow for the possibility of coefficients on the variables.

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The count is $2$ and $6$ for $p=2$ and $p=3$ respectively, and otherwise $p-1$ or $p+1$ according as $p$ is $1$ or $-1 \bmod 3$.

More generally, it is well-known that a smooth plane conic over ${\bf Z} / p {\bf Z}$ has $p+1$ points in the projective plane, so we need only subtract the number of points at infinity, which here is the number of square roots of $-3 \bmod p$. (The primes $p=2,3$ are special because $x^2+3y^2-1$ factors as $(x+y-1)^2 \bmod 2$ and $(x+1)(x-1) \bmod 3$.)

For general $x^2 - D y^2 = 1 \bmod p$ (with $p \not\mid 2D$) the count is $p - (D/p)$ where $(D/p)$ is the Legendre symbol. For $D=-3$ we can also choose between $p-1$ and $p+1$ by observing that the solutions come in triples $\{ (x,y), \frac12(-x-3y,x-y), \frac12(-x+3y,-x-y) \}$ so the count must be a multiple of $3$. This trick is possible here because $(-1+\sqrt{-3})/2$ is a cube root of unity; likewise for $D = -1$ there's a fourth root of unity $\sqrt{-1}$, and the solutions come in quadruples $\{(\pm x, \pm y), (\pm y, \mp x)\}$ so for any odd prime $p$ the number of solutions of $x^2 + y^2 \equiv 1 \bmod p$ is whichever of $p \pm 1$ is a multiple of $4$.

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You may simply rationally parametrize this curve: if $y=0$ we have 2 solutions, otherwise we may denote $x=1+ty$, and $x^2+3y^2=1$ reads as $2t+(3+t^2)y=0$. If $-3$ is a quadratic non-residue modulo $p\ne 3$, this has unique solution $y=-2t/(3+t^2)$ for each $t\ne 0$. If $-3$ is a quadratic residue, there are two non-zero values of $t$ for which there is no solution.

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Here is an alternate approach, more algebraic and less geometric. As in Noam's answer we'll consider the more general equation $x^2 - Dy^2 \equiv 1 \bmod p$. Consider the $\mathbb{F}_p$-algebra $A = \mathbb{F}_p[\sqrt{D}] \cong \mathbb{F}_p[\alpha]/(\alpha^2 - D)$, which is not necessarily a field. It has elements of the form $x + y \sqrt{D}$ and a multiplicative norm map

$$N : A \ni x + y \sqrt{D} \mapsto N(x + y \sqrt{D}) = x^2 - Dy^2 \in \mathbb{F}_p.$$

We want to compute $|N^{-1}(1)|$, which can be done as follows. Things split into a few cases, the last few of which are degenerate. In most of the cases $N$ will turn out to be surjective which gives $|N^{-1}(1)| = \frac{|A^{\times}|}{|\mathbb{F}_p^{\times}|}$ and we will compute this.

Case 1: $\gcd(D, p) = 1$ and $\left( \frac{D}{p} \right) = -1$ (which implies $p$ odd). Then $A$ is the finite field $\mathbb{F}_{p^2}$, whose multiplicative group is cyclic of order $p^2 - 1$, and the norm map $N : \mathbb{F}_{p^2} \to \mathbb{F}_p$ is given by $N(\alpha) = \alpha \overline{\alpha} = \alpha^{p+1}$. Hence if $\alpha$ is a generator of the multiplicative group of $\mathbb{F}_{p^2}$ then $\alpha^{p+1}$ has order $p-1$ and so is a generator of the multiplicative group of $\mathbb{F}_p$. It follows that

$$\left| \{ (x, y) \in \mathbb{F}_p : x^2 - Dy^2 = 1 \} \right| = \frac{p^2 - 1}{p - 1} = p + 1.$$

Case 2: $\gcd(D, p) = 1$ and $\left( \frac{D}{p} \right) = 1$ and $p$ is odd. Then $\alpha^2 - D = (\alpha - \sqrt{D})(\alpha + \sqrt{D})$ splits into a polynomial with two distinct roots which gives $A \cong \mathbb{F}_p^2$. Under this isomorphism the norm map can be identified with the multiplication map $\mathbb{F}_p^2 \ni (x, y) \mapsto xy \in \mathbb{F}_p$ which is clearly surjective. It follows that

$$\left| \{ (x, y) \in \mathbb{F}_p : x^2 - Dy^2 = 1 \} \right| = \frac{(p - 1)^2}{p - 1} = p - 1.$$

Case 3a: $p \mid D$ and $p$ is odd. Then $\alpha^2 - D \equiv \alpha^2 \bmod p$ is a polynomial with repeated roots so $A \cong \mathbb{F}_p[\alpha]/\alpha^2$ and the norm map is $N(x + y \alpha) = x^2$. Here the norm map is not surjective: its image on units is the subgroup of squares which has order $\frac{p-1}{2}$. This gives

$$\left| \{ (x, y) \in \mathbb{F}_p : x^2 = 1 \} \right| = \frac{p(p - 1)}{ \frac{p-1}{2} } = 2p$$

which of course can be obtained in a more elementary way; I'm including this case for the sake of completeness.

Case 3b: $p = 2$. Here $\alpha^2 - D$ is either $\alpha^2 \bmod 2$ or $(\alpha - 1)^2 \bmod 2$ so $A \cong \mathbb{F}_2[\varepsilon]/\varepsilon^2$. The norm map is either $N(x + y \sqrt{D}) = x^2 = x$ if $D$ is even or $N(x + y \sqrt{D}) = x^2 - y^2 = x + y$ if $D$ is odd; either way it is clearly surjective and we get

$$\left| \{ (x, y) \in \mathbb{F}_2 : x^2 - Dy^2 = 1 \} \right| = 2$$

which is even easier to obtain but again is being included for the sake of completeness.

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For any for any nonzero residues $a,b,c\not\equiv 0\pmod{p}$, the number of solutions of the congruence $ax^2+by^2\equiv c\pmod{p}$ equals $p-\left(\frac{-ab}{p}\right)$. For the proof, see my post here. Note also that this proof generalizes easily to any number of variables.

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We give yet another approach to proving the result, using nothing more advanced than properties of the Legendre symbol.

Proposition. Suppose $p$ is an odd prime and $D$ is not a multiple of $p$. Then the number of solutions of $x^2 - Dy^2 \equiv 1 \bmod p$ is $p - (D/p)$, i.e. $p-1$ if $D$ is a square mod $p$ and $p+1$ if not.

Proof: It is clear that the answer depends only on the Legendre symbol $(D/p)$, because for any nonzero $c \bmod p$ the invertible change of variables $(x,y) \to (x,cy)$ shows that the count is the same for $D$ and $c^2 \! D$. Let $N_+$, then, be the number of solutions for $(D/p) = +1$, and let $N_-$ be the number of solutions for $(D/p) = -1$.

Next observe that $N_+ + N_- = 2p$, because each $x$ that contributes $2$, $1$, or $0$ solutions to $N_+$ contributes $0$, $1$, or $2$ solutions respectively to $N_-$. So it is enough to prove the Proposition for one of $N_+$ and $N_-$.

But $N_+$ is the number of solutions mod $p$ of $1 \equiv x^2 - y^2 = (x+y) (x-y)$. Write $(r,s) = (x+y, x-y)$, so $x,y \equiv (r\pm s)/2$. Now the equation $rs \equiv 1$ has $p-1$ solutions mod $p$: each nonzero $r \bmod p$ determines uniquely the value of $s \equiv r^{-1} \bmod p$. Therefore the equation $x^2 - y^2 \equiv 1 \bmod p$ also has $p-1$ solutions, namely $\bigl(\frac12(r+r^{-1}), \frac12(r-r^{-1})\bigr)$. This proves that $N_+ = p-1$, whence it follows that $N_- = 2p - N_+ = p+1$. QED

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We give an elementary approach to the more general equation $x^2+Dy^2=1 \mod p$.

Firstly, if $p=2$ we have $2$ solutions as either $x^2=1$ or $x^2+y^2=1$ and

if $p|D$ we have $p$ values for $y$ and $x=\pm1$ giving $2p$ solutions.

The remaining case is $(D,p)=1$ with $p$ odd.

The equation $x^2+Dy^2=z^2$ can be solved by re-writing as $z^2-x^2=Dy^2$ or $AB=Dy^2$ where $z-x=A$, $z+x=B$. An easy calculation shows this has a total of $p^2$ solutions. (Consistent with the Chevalley-Warning theorem which says the number of solutions must be divisible by $p$)

However we need to remove the $z=0$ solutions. We have $2p-1$ to remove if -D is a quadratic residue and 1 if -D is a non-residue giving $(p-1)^2$ and $p^2-1$ solutions respectively.

Since $z\neq0$ has been excluded we can divide by $p-1$ to get the number of solutions to the original equation yielding $p-1$ if $-D$ is a residue and $p+1$ otherwise.

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