Knowra Pell equation Pell equation A Diophantine equation of the form x² − Dy² = 1, where D is a positive nonsquare integer. Its integer solutions are generated by powers of a fundamental unit in the quadratic field ℚ(√D).
Continued fraction : An expression formed by repeatedly nesting integer parts and reciprocals. Convergents to √D produce solutions to the Pell equation.
Diophantine equation : A polynomial equation whose solutions are sought among integers or other specified number-system elements. The Pell equation is a classic case of an equation restricted to integer solutions.
Pell equation for D = 2 : The equation x² − 2y² = 1, whose least positive solution is (3, 2). Its solutions follow powers of 3 + 2√2 and illustrate the general generation rule.
Fermat's equation : A family of Diophantine equations studied by Pierre de Fermat, including x² − Dy² = 1. Fermat posed problems equivalent to Pell equations, helping establish them as a major number-theory topic.
Convergent (continued fraction) : A rational approximation obtained by truncating a continued fraction. Certain convergents of √D yield integer pairs satisfying x² − Dy² = ±1.
Integer : A whole number, positive, negative, or zero. Both variables must be integers for Pell’s equation to pose its characteristic problem.
Pell equation for D = 3 : The equation x² − 3y² = 1, whose least positive solution is (2, 1). Its especially small fundamental solution makes the recurrence of later solutions easy to see.
Leonhard Euler : An eighteenth-century Swiss mathematician who made foundational contributions across mathematics and physics. Euler popularized the name “Pell equation,” although John Pell did not discover its central method.
Fundamental solution (Pell equation) : The least positive integer solution (x, y) of x² − Dy² = 1, with y > 0. This smallest solution generates every positive solution by taking powers of x + y√D.
Perfect square : An integer equal to the square of another integer. The parameter D must not be a perfect square for this equation to have its Pell-type structure.
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