Quadratic form representations via generalized continuants
  H. J. S. Smith proved Fermat's two-square theorem using the notion of palindromic continuants. In this paper we extend Smith's approach to proper binary quadratic form representations in some commutative Euclidean rings, including rings of integers and rings of polynomials over fields of odd characteristic. Also, we present new deterministic algorithms for finding the corresponding proper representations. 
Charles Delorme and Guillermo Pineda-Villavicencio
math.NT