Midpoint criteria for solving Pell’s equation using the nearest square continued fraction

Matthews, Keith; Robertson, John; White, Jim

doi:10.1090/S0025-5718-09-02286-8

Midpoint criteria for solving Pell’s equation using the nearest square continued fraction
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by Keith Matthews, John Robertson and Jim White PDF

Math. Comp. 79 (2010), 485-499 Request permission

Abstract:

We derive midpoint criteria for solving Pell’s equation $x^2-Dy^2=\pm 1$, using the nearest square continued fraction expansion of $\sqrt {D}$. The period of the expansion is on average $70\%$ that of the regular continued fraction. We derive similar criteria for the diophantine equation $x^2-xy-\frac {(D-1)}{4}y^2=\pm 1$, where $D\equiv 1\pmod {4}$. We also present some numerical results and conclude with a comparison of the computational performance of the regular, nearest square and nearest integer continued fraction algorithms.

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