An elementary proof of the halting property for Chakravala algorithm

TL;DR

This paper provides a simplified and correct proof that the Chakravala algorithm, used for solving Pell's equation, always terminates, clarifying a historically claimed but previously unproven property.

Contribution

The paper offers a refined, elementary proof confirming the halting property of the Chakravala algorithm, addressing a long-standing gap in its theoretical understanding.

Findings

Proof confirms the algorithm always halts

Simplifies previous complex proofs

Clarifies historical claims about the algorithm's properties

Abstract

In 1930, A.A.K. Ayyangar allegedly produced the missing proof that the ancient Indian Chakravala algorithm, designed to solve Pell's equation, always halts. Refining his own elementary arguments, we give a correct and shorter proof.