A generalisation of Pell’s equation

We want to find the solutions to

(*)\qquad x^2+mxy+ny^2=1

over the integers, where m,n\in\mathbb Z.

Let \alpha and \bar\alpha be the roots of x^2+mx+n=0. We can factorise (*) as (x-y\alpha)(x-y\bar\alpha)=1. If \alpha is rational, it must be an integer and then it is easy to find all solutions. So let us consider the more interesting case where \alpha is not rational (i.e. \sqrt{m^2-4n}\not\in\mathbb Z).

Now a solution in the form z=x-y\alpha to (*) is a unit in the ring \mathbb Z[\alpha]. For z=x-y\alpha\in\mathbb Z[\alpha], denote its conjugate by \bar z=x-y\bar\alpha. Then we have the multiplicative norm

N(z)=N(x-y\alpha)=z\bar z=x^2+mxy+ny^2.

It follows that the solutions to (*) form a subgroup S of the multiplicative group of units of \mathbb Z[\alpha].

Case 1: m^2< 4n. Then (*) defines an ellipse, so there are at most finitely many solutions, i.e. S is a finite subgroup of \mathbb C^\times. Hence S is cyclic and every solution is of the form

\displaystyle x-y\alpha=\cos\frac{2\pi t}{k}+i\sin\frac{2\pi t}{k},\quad x-y\bar\alpha=\cos\frac{2\pi t}{k}-i\sin\frac{2\pi t}{k}

\displaystyle \Rightarrow 2x+my=2\cos\frac{2\pi t}{k}\in [-2,2]\cap\mathbb Z\Rightarrow \cos\frac{2\pi t}{k}\in\{0,\pm\frac 12,\pm1\}

\displaystyle\Rightarrow x-y\alpha\in\{\pm 1,\pm i, \frac{1\pm i\sqrt 3}{2}, \frac{-1\pm i\sqrt 3}{2}\}.

Case 2: m^2>4n. Now \alpha is real, so S is a subgroup of \mathbb R^\times.

Claim. S is not dense.

Proof. WLOG S\neq\{\pm 1\}. Then for z=x-y\alpha\in S\backslash\{\pm 1\} we have

\displaystyle \left |z-\frac 1z\right |=|y(\alpha-\bar\alpha)|\ge |\alpha-\bar\alpha|=\sqrt{m^2-4n}=:\varepsilon> 1.

Suppose that 1<z<\varepsilon is a solution. Then

\displaystyle \left |z-\frac 1z\right |=z-\frac 1z<\varepsilon-\frac 1\varepsilon<\varepsilon,

impossible. Thus S\cap (1,\varepsilon)=\emptyset, i.e. S is not dense. \square

It follows that S is either \{\pm 1\} or \langle\delta\rangle\cup -\langle\delta\rangle for some \delta>1 (cf. ordering in groups, corollary 4). Therefore all solutions (x,y) to (*) are either just (\pm 1,0), or are given by

\boxed{x-y\alpha=\pm (x_0-y_0\alpha)^k,\; k\in\mathbb Z}

where x_0-y_0\alpha is the least solution >1.

Corollary. (Dirichlet’s unit theorem for a real quadratic field) If K is a real quadratic field, then \mathcal O_K^*=\{\pm\varepsilon^n: n\in\mathbb Z\} for some \varepsilon\ge 1.


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Filed under Algebra, Number theory

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