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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the computational complexity of determining the solvability or unsolvability of the equation $X^{2}-DY^{2}=-1$

Author: J. C. Lagarias
Journal: Trans. Amer. Math. Soc. 260 (1980), 485-508
MSC: Primary 10B05; Secondary 68C25
MathSciNet review: 574794
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Abstract: The problem of characterizing those D for which the Diophantine equation ${X^2} - D{Y^2} = - 1$ is solvable has been studied for two hundred years. This paper considers this problem from the viewpoint of determining the computational complexity of recognizing such D. For a given D, one can decide the solvability or unsolvability of ${X^2} - D{Y^2} = - 1$ using the ordinary continued fraction expansion of $\sqrt D$, but for certain D this requires more than $\tfrac {1}{3} \sqrt D {(\log D)^{ - 1}}$ computational operations. This paper presents a new algorithm for answering this question and proves that this algorithm always runs to completion in $O({D^{1/4 + \varepsilon }})$ bit operations. If the input to this algorithm includes a complete prime factorization of D and a quadratic nonresidue ${n_i}$ for each prime ${p_i}$ dividing D, then this algorithm is guaranteed to run to completion in $O({(\log D)^5} (\log \log D)(\log \log \log D))$ bit operations. This algorithm is based on an algorithm that finds a basis of forms for the 2-Sylow subgroup of the class group of binary quadratic forms of determinant D.

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Keywords: Computational complexity, binary quadratic forms, form class group, composition of forms, Pell’s equation, Diophantine equation
Article copyright: © Copyright 1980 American Mathematical Society