Kuṭṭaka, Bhāvanā and Cakravāla

Ancient Indian mathematical treatises contain ingenious methods for finding integer solutions of indeterminate (or Diophantine) equations. The three greatest landmarks in this area are the kuṭṭaka method of Āryabhaṭa for solving the linear indeterminate equation ay − bx c c, the bhāvanā law of Brahmagupta, and the cakravāla algorithm described by Jayadeva and Bhāskara II for solving the quadratic indeterminate equation Dx2 + 1 = y2. We shall briefly recall the history of the above equations in ancient India and of their rediscovery in Europe, give an account of the ancient Indian texts dealing with algebra in general and the above indeterminate equations in particular, and mention a few works on history of mathematics which have highlighted ancient Indian algebra, especially indeterminate equations. We shall then discuss various mathematical aspects of the kuṭṭaka, bhāvanā and cakravāla from the viewpoint of modern algebra and number theory and the general cultural atmosphere in which the leading Indian mathematicians undertook such explorations. We shall also examine the coverage of these results in texts involving history of mathematics.