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Pell Equations: Non-Principal Lagrange Criteria and Central Norms

  • R. A. Mollin (a1) and A. Srinivasan (a2)

Abstract

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 $D$ .

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References

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[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, 485508.
[2] Lenstra, H. W. Jr., Solving the Pell equation. Notices Amer. Math. Soc. 49(2002), no. 2, 182192.
[3] Mollin, R. A., Quadratics. CRC Press Series on Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, 1996.
[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, 159207.
[5] Mollin, R. A., Lagrange, central norms, and quadratic Diophantine equations. Int. J. Math. Math. Sci. 2005, no. 7, 10391047.
[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, 121132. http://dx.doi.org/10.4153/CMB-2005-011-0
[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.
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Pell Equations: Non-Principal Lagrange Criteria and Central Norms

  • R. A. Mollin (a1) and A. Srinivasan (a2)

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