Why is it called the Pell equation?

Through a historical error: Euler attributed the equation to John Pell, though Pell did little work on it; the substantial early advances were made by Indian mathematicians and by Fermat and Lagrange.

How do you find a Pell solution?

Expand the square root of d as a continued fraction; its periodic convergents yield the fundamental solution, from which every other solution is generated by repeated composition.

Linear and Pell Equations

Linear Diophantine equations are solved completely by the Euclidean algorithm, while the Pell equation, asking for integer solutions of x squared minus d y squared equals one, reveals the deep structure of real quadratic fields through continued fractions.

Definition

A linear Diophantine equation seeks integer solutions of a linear equation with integer coefficients; the Pell equation is the quadratic Diophantine equation x squared minus d y squared equals one for a nonsquare positive integer d, whose solutions form an infinite, finitely generated family.

Scope

This topic covers linear Diophantine equations in two or more variables and their full solution via greatest common divisors and Bezout's identity, the Pell equation and its negative and generalized forms, the continued fraction expansion of quadratic irrationals, the fundamental solution and how all solutions are generated from it, and the connection to the units and fundamental unit of a real quadratic field.

Core questions

When does a linear Diophantine equation have integer solutions, and how is the full solution set described?
Why does the Pell equation always have nontrivial solutions for nonsquare d?
How does the continued fraction expansion of the square root of d produce the fundamental solution?
How are all Pell solutions generated from the fundamental one, and how does this relate to units of a quadratic field?

Key theories

Solvability of linear Diophantine equations: The equation a x plus b y equals c has integer solutions exactly when the greatest common divisor of a and b divides c, and Bezout's identity then gives a particular solution and the full one-parameter family.
Existence and structure of Pell solutions: For nonsquare d the Pell equation has infinitely many solutions; a fundamental solution exists, and all others are obtained by taking powers of the corresponding unit in the real quadratic field.
Continued fractions and quadratic irrationals: The continued fraction expansion of the square root of d is eventually periodic, and its convergents furnish the fundamental Pell solution, tying Diophantine solvability to Diophantine approximation.

Clinical relevance

Pell-type equations and continued fractions appear in algorithms for computing fundamental units and regulators of quadratic fields and in approximating irrational ratios, with practical use in calendar design, gear ratios, and lattice reduction.

History

Indian mathematicians, notably Brahmagupta in the seventh century and Bhaskara II with the chakravala method, solved Pell's equation centuries before Europe. Fermat posed it as a challenge, and Lagrange gave the first complete European proof in 1768; the name Pell is a historical misattribution by Euler.

Key figures

Brahmagupta
Joseph-Louis Lagrange
Pierre de Fermat
John Pell

Seminal works

hardyWright2008

Frequently asked questions

Why is it called the Pell equation?: Through a historical error: Euler attributed the equation to John Pell, though Pell did little work on it; the substantial early advances were made by Indian mathematicians and by Fermat and Lagrange.
How do you find a Pell solution?: Expand the square root of d as a continued fraction; its periodic convergents yield the fundamental solution, from which every other solution is generated by repeated composition.