Vargaprakrti: The 'Pell Equation' 1,000 Years Before Pell

India's solution to one of mathematics' hardest problems

Discover the chakravala (cyclic method) for solving indeterminate quadratic equations, Bhaskara II's elegant solution predating Fermat and Pell by centuries.

Fermat's Challenge to England

In 1657, in his study in Toulouse, the French magistrate Pierre de Fermat dipped his pen and wrote a letter to the mathematicians of England with a challenge. Find positive integer solutions to the equation 61x² + 1 = y². He added, with a touch of malice, that he was not interested in the easy cases like x = 1 that work for small coefficients. He wanted real solutions, and he wanted them for a number, 61, that he had chosen because he knew the answer would be enormous. The smallest positive integer x that satisfies 61x² + 1 = y² is 226,153,980. The corresponding y is 1,766,319,049. Fermat already knew this. He wanted to see whether Europe could find it.

Europe could not. John Wallis and William Brouncker in England produced a method that worked in principle but missed the elegance Fermat was looking for. A century later, in 1768, Joseph-Louis Lagrange finally gave a systematic proof that solutions exist for every such equation. By then the problem had been named after the wrong man. In 1732 Leonhard Euler, reading Wallis's account of the Fermat-Brouncker correspondence, attributed the equation in passing to John Pell, an English mathematician whose actual contribution to the problem was essentially nothing. The name stuck. Every textbook on number theory still calls it the Pell equation.

The problem has a different name in Sanskrit. It is called vargaprakṛti, literally 'square-nature', the study of equations where a square quantity enters as the essential structure of the relationship. And by the time Fermat wrote his challenge, vargaprakṛti had been a solved topic in Indian mathematics for more than a thousand years. The cyclic method that closes it, the chakravāla, was already complete in Bhāskara II's Bījagaṇita, written in 1150 CE. The specific case Fermat chose, 61x² + 1 = y², was already worked out as an example in that same text, with the full answer Fermat thought nobody could find.