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H. J. S. Smith proved Fermat's two-square theorem using the notion ofpalindromic continuants. In this paper we extend Smith's approach to properbinary quadratic form representations in some commutative Euclidean rings,including rings of integers and rings of polynomials over fields of oddcharacteristic. Also, we present new deterministic algorithms for finding thecorresponding proper representations.

Quadratic form representations via generalized continuants