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Solving the Pell equation. (English) Zbl 1043.11027
Bennett, M. A. (ed.) et al., Number theory for the millennium III. Proceedings of the millennial conference on number theory, Urbana-Champaign, IL, USA, May 21–26, 2000. Natick, MA: A K Peters (ISBN 1-56881-152-7/hbk). 397-435 (2002).
While the Diophantine equation ${x}^{2}-D{y}^{2}=1$ ($D$ a nonsquare positive integer parameter) was spuriously named in honour of the seventeenth century Englishman, John Pell, it was seriously treated in India during the first millenium (CE) and in Europe from the time of Fermat. In the eighteenth century, Lagrange showed how the continued fraction respresentation of $\sqrt{D}$ could be used to obtain solutions. After reviewing this history, the author provides a comprehensive survey of recent algebraic and computational research on obtaining solutions and concludes with some open questions to indicate that there is still significant work to be done. Because the basic solution can be large relative to $D$, it is convenient to suppose that $D$ is squarefree and to study the regulator $R=logϵ$, where $ϵ$ is the fundamental unit of a quadratic field derived from $D$, a small power of which yields a solution of Pell’s equation. Through a formula that relates $R$ to the class number of the field and a certain $L$-series, it is possible to obtain estimates for $R$. Assuming the generalized Riemann Hypothesis, one obtains a subexponential algorithm for computing $R$ that dates back to 1989 and has been refined since.
##### MSC:
 11D09 Quadratic and bilinear diophantine equations 11-02 Research monographs (number theory) 11Y50 Computer solution of Diophantine equations