Pell's equation has connections to several other important subjects in mathematics.

The general form of Pell's equation using the chakravala method.

He wrote on the link between continued fractions and Pell's equation.

He was the first European to solve what is now known as Pell's equation.

The problem of finding square triangular numbers reduces to Pell's equation in the following way.

This equation is different in form from Pell's equation but equivalent to it.

Equations of the second form are named Pell's equations.

The theory for real quadratic fields is essentially the theory of Pell's equation.

The problem of determining, given two such sets, all numbers that belong to both can be solved by reducing the problem to Pell's equation.

Important special cases of continued fraction convergents include the Fibonacci numbers and solutions to Pell's equation.

The resulting algorithm for solving Pell's equation is more efficient than the continued fraction method, though it still does not take polynomial time.

Products of such matrices take exactly the same form, and thus all such products yield solutions to Pell's equation.

The relationship to the continued fractions implies that the solutions to Pell's equation form a semigroup subset of the modular group.

Such almost-isosceles right-angled triangles can be obtained recursively using Pell's equation:

Pell's equation, first misnamed by Euler.

Pell's equations were studied as early as 400 BC in India and Greece.

Demeyer (2007) mentions a connection between Pell's equation and the Chebyshev polynomials:

As Lenstra (2002) describes, Pell's equation can also be used to solve Archimedes' cattle problem.

Pell's equation and the Pell number are both named after 17th century mathematician John Pell.

The theory of Pell's equation may be viewed as a part of the theory of binary quadratic forms.

He then used this lemma to both generate infinitely many (integral) solutions of Pell's equation, given one solution, and state the following theorem:

The general theory of Pell's equation, based on continued fractions and algebraic manipulations with numbers of the form was developed by Lagrange in 1766-1769.

Thus, the fundamental solution may be found by performing the continued fraction expansion and testing each successive convergent until a solution to Pell's equation is found.

The geometry can be worked independently of knowledge of Pell's equation or of the properties of convergents of a continued fraction, but to similar effect.

Theorem (Brahmagupta): If the equation has an integer solution for any one of then Pell's equation: