On the Brahmagupta-Fermat-Pell Equation: The Chakravāla or Cyclic algorithm revisited

Pronob Mitter (1)

1 L2C - Laboratoire Charles Coulomb

In the following pages we take a fresh look at the ancient Indian Chakravāla or Cyclic algorithm for solving the Brahmagupta-Fermat-Pell quadratic Diophantine equation in integers taking account of recent developments. This is the oldest general algorithm (1150 CE) for solving this equation. The algorithm can be proved directly in its own terms, following a recent work by A. Bauval, to always lead to a periodic solution in a finite number of steps. We review a slightly modified version of this work. It forms the basis of a reinterpretation of this algorithm in the framework of a reduction theory of binary indefinite integer valued quadratic forms on which the modular group SL(2, Z) acts. The reduction condition of this algorithm are restated in terms of the roots of the quadratic form. The SL(2, Z) action on (reduced) roots of this form furnish semi-regular continued fractions which are periodic and which furnish SL(2, Z) automorphisms of the quadratic form. Very much as in the classical theory of Gauss, this gives solutions of the Brahmagupta-Fermat-Pell equation. We give a proof that the solution at the end of the first cycle is fundamental (positive and least). We also give the conversion to regular continued fractions which involve larger periods. A number of worked out examples are given to illustrate the main points and this for the benefit of the uninitiated reader.

Mots clés: Brahmagupta; Bhaskara; Cyclic; Fermat; Diophantine Equation

Domaines: Mathématiques [math]

Pronob Mitter. On the Brahmagupta-Fermat-Pell Equation: The Chakravāla or Cyclic algorithm revisited. 2023. ⟨hal-04173618⟩