matqkks said:What is the best way to introduce Pell’s equation on a first elementary number theory course? Are there any practical applications of Pell’s equation? What are the really interesting questions about Pell’s equation? Are there any good resources on Pell’s equation.
Pell's equation is a mathematical equation of the form x^2 - Dy^2 = 1, where D is a positive non-square integer. It is named after the English mathematician John Pell, who studied its properties in the 17th century.
Pell's equation has many applications in number theory, algebra, and cryptography. It has also been used to solve various real-world problems, such as finding the best approximation for square roots and calculating the shortest distance between two lattice points in a plane.
There are many books, websites, and online courses available for learning about Pell's equation. Some recommended resources include "An Introduction to Pell's Equation" by Edward J. Barbeau, "The Pell Equation" by Edward Burger and James L. Harkness, and the lecture series "Pell's Equation and Continued Fractions" by Keith Conrad.
Yes, there are various methods for solving Pell's equation, including the Chakravala method, the Lagrange method, and continued fractions. Each method has its own advantages and can be applied to different types of Pell's equations.
No, Pell's equation can only be solved for certain values of D. These values are known as fundamental solutions, and they can be found using the continued fraction method. For other values of D, there may be no solutions or an infinite number of solutions.