Pell's Equation - History

History

Pell's equations were studied as early as 400 BC in India and Greece. The Indian and Greek mathematicians were mainly interested in the n = 2 case of Pell's equation,

because of its connection to the square root of two. Indeed, if x and y are positive integers satisfying this equation, then x/y is an approximation of √2. For example, Baudhayana discovered that x = 17, y = 12 and x = 577, y = 408 are two solutions to the Pell equation, and that 17/12 and 577/408 are very close approximations to the square root of two.

Later, Archimedes used a similar equation to approximate the square root of 3 by the rational number 1351/780.

Around AD 250, Diophantus considered the equation

where a and c are fixed numbers and x and y are the variables to be solved for. This equation is different in form from Pell's equation but equivalent to it. Diophantus solved the equation for (a,c) equal to (1,1), (1,−1), (1,12), and (3,9). Al-Karaji, a 10th-century Persian mathematician, worked on similar problems to Diophantus.

In Indian mathematics, Brahmagupta discovered that

(see Brahmagupta's identity). Using this, he was able to "compose" triples and that were solutions of, to generate the new triple

and

Not only did this give a way to generate infinitely many solutions to starting with one solution, but also, by dividing such a composition by, integer or "nearly integer" solutions could often be obtained. For instance, for, Brahmagupta composed the triple (since ) with itself to get the new triple . Dividing throughout by 64 gave the triple, which when composed with itself gave the desired integer solution . Brahmagupta solved many Pell equations with this method; in particular he showed how to obtain solutions starting from an integer solution of for k=±1, ±2, or ±4.

The first general method for solving the Pell equation (for all N) was given by Bhaskara II in 1150, extending the methods of Brahmagupta. Called the chakravala (cyclic) method, it starts by composing any triple (that is, one which satisfies ) with the trivial triple to get the triple, which can be scaled down to

When m is chosen so that (a+bm)/k is an integer, so are the other two numbers in the triple. Among such m, the method chooses one that minimizes (m²-N)/k, and repeats the process. This method always terminates with a solution (proved by Lagrange in 1768). Bhaskara used it to give the solution x=1766319049, y=226153980 to the notorious N=61 case.

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.