The 'Pell' equation and the chakravala method
AI-drafted · awaiting admin review
Brahmagupta (628 CE), Jayadeva and Bhaskara II (1150) solved equations of the form Nx² + 1 = y², including the hard case N = 61, about 500 years before Fermat posed it in Europe in 1657.
India first (well documented)Mathematics & astronomy
Indian claim
Indian mathematicians solved the so-called Pell equation centuries before Europe, with the chakravala method.
Evidence for the Indian claim
- Brahmagupta (628 CE) found the composition rule (bhavana) that builds new solutions from old ones.
- Jayadeva (11th century or earlier) and Bhaskara II (Bijaganita, 1150) gave the cyclic chakravala method that solves any case.
- Bhaskara II solved 61x² + 1 = y², whose smallest answer is x = 226,153,980 and y = 1,766,319,049.
Commonly credited to
Fermat posed the problem in 1657, Brouncker and Wallis solved cases, and Lagrange gave the complete theory; Euler wrongly named it after John Pell.
Evidence for the other claim
- Fermat challenged European mathematicians with the same N = 61 case in 1657.
- Lagrange proved in the 1760s that the equation always has a solution; the Indian texts give the method but not this proof.
- Historian Clas-Olof Selenius (1975) judged the chakravala a highly efficient algorithm, different from the later European methods.
What historians and scientists say
Historians fully credit India with solving this equation first; some writers now call it the Brahmagupta–Bhaskara equation. Europe added the general proof.