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Pell Equations: Non-Principal Lagrange Criteria and Central Norms

Published online by Cambridge University Press:  20 November 2018

R. A. Mollin  and
A. Srinivasan
R. A. Mollin
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, ABURL: http://www.math.ucalgary.ca/~ramollin/e-mail: ramollin@math.ucalgary.ca
A. Srinivasan
Affiliation:
Department of Mathematics, Siddhartha college, (affiliated with Mumbai University), Indiae-mail: rsrinivasan.anitha@gmail.com
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Abstract

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We provide a criterion for the central norm to be any value in the simple continued fraction expansion of $\sqrt{D}$ for any non-square integer $D\,>\,1$. We also provide a simple criterion for the solvability of the Pell equation ${{x}^{2}}\,-\,D{{y}^{2}}\,=\,-1$ in terms of congruence conditions modulo $D$.

Keywords

11D0911A5511R1111R29Pell's equationcontinued fractionscentral norms
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 4 , 01 December 2012 , pp. 774 - 782
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Lagarias, J. C., On the computational complexity of determining the solvability or unsolvability of the equation X 2 – DY 2 = –1. Trans. Amer. Math. Soc. 260(1980), no. 2, 485–508.Google Scholar
[2] Lenstra, H. W. Jr., Solving the Pell equation. Notices Amer. Math. Soc. 49(2002), no. 2, 182–192.Google Scholar
[3] Mollin, R. A., Quadratics. CRC Press Series on Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, 1996.Google Scholar
[4] Mollin, R. A., A continued fraction approach to the Diophantine equation ax 2 – by 2 = 1. JP J. Algebra Number Theory Appl. 4(2004), no. 1, 159–207.Google Scholar
[5] Mollin, R. A., Lagrange, central norms, and quadratic Diophantine equations. Int. J. Math. Math. Sci. 2005, no. 7, 1039–1047.Google Scholar
[6] Mollin, R. A., Necessary and sufficient conditions for the central norm to equal 2 h in the simple continued fraction expansion of for any odd c > 1. Canad. Math. Bull. 48(2005), no. 1, 121–132. http://dx.doi.org/10.4153/CMB-2005-011-0 +1.+Canad.+Math.+Bull.+48(2005),+no.+1,+121–132.+http://dx.doi.org/10.4153/CMB-2005-011-0>Google Scholar
[7] Mollin, R. A., Fundamental number theory with applications. Second ed., Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2008.Google Scholar