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.References
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Additional Information
- Keith Matthews
- Affiliation: Department of Mathematics, University of Queensland, Brisbane, Australia, 4072 and Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia
- Email: keithmatt@gmail.com
- John Robertson
- Affiliation: Actuarial and Economic Services Division, National Council on Compensation Insurance, Boca Raton, Florida 33487
- Email: jpr2718@gmail.com
- Jim White
- Affiliation: Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia
- Email: mathimagics@yahoo.co.uk
- Received by editor(s): July 29, 2008
- Received by editor(s) in revised form: March 15, 2009
- Published electronically: July 21, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 79 (2010), 485-499
- MSC (2000): Primary 11D09, 11Y50, 11A55, 11Y65
- DOI: https://doi.org/10.1090/S0025-5718-09-02286-8
- MathSciNet review: 2552236