Pell's equation
Introduction
Pell's equation, named after the English mathematician John Pell, is a specific type of Diophantine equation in number theory. Diophantine equations are polynomial equations, usually in two or more unknowns, that are defined by requiring integer solutions. The equation is of the form x^2 - ny^2 = 1, where n is a given positive nonsquare integer and x and y are variables which represent integers.
History
The equation is named after John Pell, who lived in the 17th century, but it was studied by ancient mathematicians much before Pell's time. The Indian mathematician Brahmagupta was one of the first to provide a solution for the equation in the 7th century. The equation was also studied by the French mathematician Fermat in the 17th century. Despite the name, Pell did not contribute significantly to the study of this equation.
Mathematical Background
Pell's equation is a special case of a more general class of equations known as quadratic Diophantine equations. These equations are a subset of the larger field of Diophantine equations, named after the ancient Greek mathematician Diophantus who made significant contributions to the study of these equations.
Solutions
The solutions to Pell's equation can be found using a variety of methods. One common approach is to use the method of continued fractions. This method involves expressing the square root of n as a continued fraction, then using the convergents of this fraction to generate solutions to the equation.
Another method for finding solutions to Pell's equation involves the use of algebraic number theory, specifically the theory of quadratic fields. This approach involves viewing the equation in the context of the ring of integers of a quadratic field, and using properties of this ring to find solutions.
Applications
Pell's equation has many applications in number theory and other areas of mathematics. For example, it plays a key role in the solution of the Archimedean cattle problem, a problem posed by Archimedes that involves finding the number of cattle in a herd given certain conditions.
In addition, Pell's equation is used in the study of quadratic forms, which are polynomials with terms of degree two. The equation also has applications in the field of cryptography, particularly in the area of public key cryptography.