About This Calculator

1. What is this calculator for?
2. Can I embed this on my website?
3. How do I solve the Pell equation using Bhaskara II's method?

What is this calculator for?

This calculator solves the Pell equation x^2 - Dy^2 = 1 for a given non-square positive integer D. It finds the fundamental solution (x, y) and can generate further solutions.

Can I embed this on my website?

Sure. Embedding is allowed as long as you promise to follow our conditions. Here's the embed code:

<iframe width="415" height="220" src="http://www.a-calculator.com/pell-equation/embed.html" frameborder="0" allowtransparency="true"></iframe>

How do I solve the Pell equation using Bhaskara II's method?

The Pell equation is a Diophantine equation of the form \(x^2 - Dy^2 = 1\), where \(D\) is a positive non-square integer, and \(x\) and \(y\) are integers to be found. Bhaskara II's method for solving this equation involves using continued fractions to find the fundamental solution. Here's an overview of the process:

1. Find the Continued Fraction Expansion of \(\sqrt{D}\): Express the square root of \(D\) as a continued fraction. This continued fraction will be periodic and contain the information needed to find the solutions to the Pell equation.
2. Identify the Period: The period of the continued fraction expansion of \(\sqrt{D}\) is the sequence of numbers that repeats indefinitely. The length of the period can vary depending on \(D\).
3. Find the Convergents: The convergents of the continued fraction are the fractions that best approximate \(\sqrt{D}\). They are obtained by truncating the continued fraction expansion at various points. The convergents are of the form \(\frac{h_n}{k_n}\), where \(h_n\) and \(k_n\) are integers.
4. Identify the Fundamental Solution: The fundamental solution to the Pell equation is the convergent \(\frac{h_n}{k_n}\) just before the period of the continued fraction repeats. If the period of the continued fraction is of length \(m\), then the fundamental solution is given by the \(m\)th convergent, \(\frac{h_m}{k_m}\), such that \(h_m^2 - Dk_m^2 = \pm 1\). The sign depends on the length of the period; if the period is even, the equation is \(h_m^2 - Dk_m^2 = -1\), and if it's odd, the equation is \(h_m^2 - Dk_m^2 = 1\).
5. Generate Further Solutions: Once the fundamental solution \((x_1, y_1) = (h_m, k_m)\) is found, all other solutions can be generated using the recurrence relations: \[ x_{n+1} = x_1 x_n + Dy_1 y_n, \quad y_{n+1} = x_1 y_n + y_1 x_n \] These relations give an infinite sequence of solutions \((x_n, y_n)\) to the Pell equation.

Bhaskara II's method is elegant and efficient, especially for finding the fundamental solution to the Pell equation. Once the fundamental solution is known, generating additional solutions is straightforward using the recurrence relations.