Diophantine Equations — Pell's Equation and Thue-Mahler

💡 The Core Insight

Pell's equation $x^2 - Dy^2 = 1$ has infinitely many solutions governed by the fundamental solution and forms a group under a natural operation, while Thue-Mahler equations $F(x,y) = m$ with $\deg(F) \geq 3$ have only finitely many solutions—a dichotomy that reflects the deep connection between Diophantine equations and algebraic geometry through the genus of associated curves.

Definition

Pell's Equation: For a positive integer $D$ that is not a perfect square, Pell's equation is

x^2 - Dy^2 = 1

where we seek integer solutions $(x, y) \in \mathbb{Z}^2$ .

Negative Pell's Equation: The equation $x^2 - Dy^2 = -1$ .

Thue Equation: For an irreducible binary form $F(x,y) \in \mathbb{Z}[x,y]$ of degree $n \geq 3$ and a non-zero integer $m$ , the Thue equation is

F(x,y) = m

where we seek integer solutions $(x,y) \in \mathbb{Z}^2$ .

Thue-Mahler Equation: Let $F(x,y) \in \mathbb{Z}[x,y]$ be an irreducible binary form of degree $n \geq 3$ , let $S = \{p_1, \ldots, p_s\}$ be a finite set of primes, and let $m \in \mathbb{Z} \setminus \{0\}$ . The Thue-Mahler equation is

F(x,y) = m \prod_{i=1}^{s} p_i^{z_i}

where we seek solutions $(x,y,z_1,\ldots,z_s) \in \mathbb{Z}^{2+s}$ with $\gcd(x,y) = 1$ .

TheoremMain Results

Theorem 1 (Existence of Solutions to Pell's Equation): Let $D$ be a positive integer that is not a perfect square. Then the equation $x^2 - Dy^2 = 1$ has infinitely many solutions in positive integers.

Theorem 2 (Fundamental Solution): Among all positive integer solutions to $x^2 - Dy^2 = 1$ , there exists a unique solution $(x_1, y_1)$ with minimal $x_1 > 1$ (equivalently, minimal $y_1 > 0$ ), called the fundamental solution. Every other positive solution $(x_n, y_n)$ is given by

x_n + y_n\sqrt{D} = (x_1 + y_1\sqrt{D})^n

for $n \in \mathbb{N}$ .

Theorem 3 (Thue's Theorem): Let $F(x,y) \in \mathbb{Z}[x,y]$ be an irreducible binary form of degree $n \geq 3$ , and let $m \in \mathbb{Z} \setminus \{0\}$ . Then the equation $F(x,y) = m$ has only finitely many solutions $(x,y) \in \mathbb{Z}^2$ .

Theorem 4 (Thue-Mahler Finiteness): Let $F(x,y) \in \mathbb{Z}[x,y]$ be an irreducible binary form of degree $n \geq 3$ , let $S$ be a finite set of primes, and let $m \in \mathbb{Z} \setminus \{0\}$ . Then the Thue-Mahler equation $F(x,y) = m \prod_{p \in S} p^{z_p}$ has only finitely many solutions with $\gcd(x,y) = 1$ .

Proof

Proof of Theorem 1 and 2: We prove these together using the theory of continued fractions and algebraic units. Step 1: Connection to units in $\mathbb{Z}[\sqrt{D}]$ . Note that if $(x,y)$ is a solution to $x^2 - Dy^2 = 1$ , then

N(x + y\sqrt{D}) = (x + y\sqrt{D})(x - y\sqrt{D}) = x^2 - Dy^2 = 1

where $N$ is the norm from $\mathbb{Q}(\sqrt{D})$ to $\mathbb{Q}$ . Thus $x + y\sqrt{D}$ is a unit in $\mathbb{Z}[\sqrt{D}]$ . Conversely, any unit $u = x + y\sqrt{D} \in \mathbb{Z}[\sqrt{D}]$ with $N(u) = 1$ gives a solution. Step 2: Existence via continued fractions. The continued fraction expansion of $\sqrt{D}$ is eventually periodic:

\sqrt{D} = [a_0; \overline{a_1, a_2, \ldots, a_r}]

Let $p_k/q_k$ denote the $k$ -th convergent. A classical result states that if the period length is $r$ , then

p_{r-1}^2 - Dq_{r-1}^2 = (-1)^r

This gives either a solution to Pell's equation (if $r$ is even) or to the negative Pell equation (if $r$ is odd). In the latter case, $(p_{2r-1}, q_{2r-1})$ satisfies $p_{2r-1}^2 - Dq_{2r-1}^2 = 1$ . Step 3: Fundamental solution. Let $(x_1, y_1)$ be the positive solution with minimal $y_1 > 0$ (equivalently, minimal $x_1 > 1$ ). Set $\varepsilon = x_1 + y_1\sqrt{D} > 1$ . This is the fundamental unit of $\mathbb{Z}[\sqrt{D}]$ among those with norm 1. Step 4: All solutions from powers. For any $n \in \mathbb{Z}$ , $\varepsilon^n$ has norm 1. Write $\varepsilon^n = x_n + y_n\sqrt{D}$ where $x_n, y_n \in \mathbb{Z}$ . Then $(x_n, y_n)$ satisfies Pell's equation. Conversely, suppose $(x,y)$ is any positive solution. Then $u = x + y\sqrt{D}$ is a unit with $u > 1$ . We can write $u = \varepsilon^n$ for some $n \in \mathbb{Z}$ (since the units of norm 1 form a cyclic group generated by $\varepsilon$ ). Since $u > 1$ and $\varepsilon > 1$ , we have $n \geq 1$ . The formulas for $x_n, y_n$ follow from the binomial expansion and the recurrence relations:

x_{n+1} = x_1 x_n + Dy_1 y_n, \quad y_{n+1} = x_1 y_n + y_1 x_n

This establishes infinitely many distinct solutions and their parametrization. □

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Proof

Proof Sketch of Theorem 3 (Thue's Theorem): We provide an outline of the classical approach. Step 1: Reduction to approximation problem. Write $F(x,y) = a_n \prod_{i=1}^{n}(x - \alpha_i y)$ where $\alpha_i$ are the (distinct) roots of $F(x,1)$ in $\mathbb{C}$ . If $F(x,y) = m$ with $y \neq 0$ , dividing by $y^n$ gives

m/y^n = a_n \prod_{i=1}^{n}(x/y - \alpha_i)

For at least one root $\alpha$ , we must have

|x/y - \alpha| \leq C|y|^{-n+1}

for some constant $C$ depending on $F$ and $m$ . Step 2: Diophantine approximation. Since $F$ is irreducible of degree $n \geq 3$ , at least one root $\alpha$ has degree $n$ over $\mathbb{Q}$ . The inequality above shows $x/y$ approximates $\alpha$ better than $C|y|^{-n+1}$ . Step 3: Application of Thue-Siegel-Roth. For any algebraic number $\alpha$ of degree $n \geq 2$ and any $\varepsilon > 0$ , the inequality

\left|\alpha - \frac{p}{q}\right| < \frac{1}{q^{n/2 + 1 + \varepsilon}}

has only finitely many solutions in coprime integers $p, q$ (this is Thue's theorem on Diophantine approximation, later strengthened by Siegel and Roth). Since $n \geq 3$ , we have $n - 1 \geq 2 > n/2 + 1$ for small $\varepsilon$ , so the approximations forced by solutions to $F(x,y) = m$ contradict Thue's approximation theorem for all but finitely many $(x,y)$ . □

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⚠️ Common mistake

Common Misconception: Not all values of $D$ admit solutions to the negative Pell equation $x^2 - Dy^2 = -1$ . For example, $D = 3$ has the solution $(1,1)$ , but $D = 3 \cdot 5 = 15$ has no solutions. The negative Pell equation is solvable if and only if the period length of the continued fraction of $\sqrt{D}$ is odd. Additionally, students often forget that Thue's finiteness theorem says nothing about how to find the solutions—it is an existence result, not a constructive one. Effective bounds came much later through Baker's work.

mathematician

Axel Thue (1909) — Proved that irreducible Diophantine equations $F(x,y) = m$ with $\deg(F) \geq 3$ have only finitely many integer solutions, pioneering the use of Diophantine approximation in solving Diophantine equations.

visualization

Example1

Solving Pell's Equation for $D = 2$ Find all positive integer solutions to $x^2 - 2y^2 = 1$ . Solution: The continued fraction expansion of $\sqrt{2}$ is $[1; \overline{2}]$ , which has period 1 (odd). Computing convergents:

$p_0/q_0 = 1/1$
$p_1/q_1 = 1 + 1/(2) = 3/2$

Since the period is odd, $(p_0, q_0) = (1,1)$ satisfies the negative Pell equation: $1^2 - 2(1)^2 = -1$ . For the positive Pell equation, we compute $(p_1, q_1)$ : indeed, $3^2 - 2(2)^2 = 9 - 8 = 1$ . The fundamental solution is $(x_1, y_1) = (3, 2)$ , with $\varepsilon = 3 + 2\sqrt{2}$ . All solutions come from:

(x_n + y_n\sqrt{2}) = (3 + 2\sqrt{2})^n

For $n=2$ :

(3 + 2\sqrt{2})^2 = 9 + 12\sqrt{2} + 8 = 17 + 12\sqrt{2}

So $(x_2, y_2) = (17, 12)$ . Verify: $17^2 - 2(12^2) = 289 - 288 = 1$ . ✓ For $n=3$ :

(3 + 2\sqrt{2})^3 = (3 + 2\sqrt{2})(17 + 12\sqrt{2}) = 51 + 36\sqrt{2} + 34\sqrt{2} + 48 = 99 + 70\sqrt{2}

So $(x_3, y_3) = (99, 70)$ . □

Example2

A Thue Equation Consider the equation $x^3 - 2y^3 = 1$ . This is a Thue equation with $F(x,y) = x^3 - 2y^3$ , which is irreducible over $\mathbb{Q}$ (by Eisenstein's criterion with $p = 2$ applied to $F(x, 1) = x^3 - 2$ ). By inspection, $(x,y) = (1, 0)$ is a solution. We can verify that $(x,y) = (-1, -1)$ is also a solution: $(-1)^3 - 2(-1)^3 = -1 + 2 = 1$ . ✓ By Thue's theorem, there are only finitely many solutions. In fact, these are the only integer solutions (this can be verified through various methods, including reduction modulo small primes and height bounds). Note the stark contrast with Pell's equation: degree 2 gives infinitely many solutions, degree 3 gives finitely many. □

Counterexample

Why Degree Matters: The Quadratic Case Consider $F(x,y) = x^2 - 2y^2$ , a binary quadratic form. The equation $x^2 - 2y^2 = 1$ is Pell's equation and has infinitely many solutions. Thue's theorem requires $\deg(F) \geq 3$ . This is not a technicality—the conclusion fails for quadratic forms. The geometric reason is that $x^2 - Dy^2 = m$ defines a genus 0 curve (a conic), which has either no rational points or infinitely many. In contrast, $F(x,y) = m$ with $\deg(F) = n \geq 3$ typically defines a curve of genus $g \geq 1$ . For example, $x^3 - 2y^3 = 1$ can be transformed into a plane cubic curve. By Faltings' theorem (formerly Mordell's conjecture), curves of genus $g \geq 2$ have only finitely many rational points, and genus 1 curves (elliptic curves) have finitely many points if the Mordell-Weil group is finite. The hypothesis $\deg(F) \geq 3$ is sharp and essential. □

🧠 Intuition

Geometric Picture: Pell's equation $x^2 - Dy^2 = 1$ describes a hyperbola in the $(x,y)$ -plane. Integer points on this hyperbola correspond to units in $\mathbb{Z}[\sqrt{D}]$ . The fundamental solution is the "smallest" unit greater than 1, and all other solutions are obtained by repeatedly "multiplying" this unit by itself—geometrically, this corresponds to a discrete but infinite sequence of points marching off to infinity along each branch of the hyperbola. For Thue equations with degree $\geq 3$ , the curve $F(x,y) = m$ is no longer a conic but a higher-degree algebraic curve. These curves "bend" more severely, and the spacing between integer points grows rapidly. The finiteness result reflects a deep principle: higher-genus curves are "more rigid" and cannot support infinitely many rational (or integer) points scattered along them. Think of it as the curve having "too much curvature" to admit infinitely many lattice points. □

connection

Connection to Cryptography: Pell's equation has surprising applications in public-key cryptography. The infrastructure of the fundamental unit in real quadratic fields, based on Pell's equation solutions, has been used to construct cryptographic protocols analogous to those based on elliptic curves. The Buchmann-Williams cryptosystem uses the arithmetic of ideal classes in real quadratic orders, where computing the fundamental solution to Pell's equation becomes a crucial subroutine. The difficulty of computing the fundamental solution for large $D$ (the regulator problem) provides potential computational hardness for cryptographic security, offering an alternative to factoring and discrete logarithm problems.

Diophantine Equations — Pell's Equation and Thue-Mahler

Diophantine Equations — Pell's Equation and Thue-Mahler

Overview

Core Theory

Examples and Applications

Historical Development

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