On Ramanujan's continued fraction
K. Ramanathan (1984)
Acta Arithmetica
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K. Ramanathan (1984)
Acta Arithmetica
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Mollin, R. (2001)
Serdica Mathematical Journal
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We investigate infinite families of integral quadratic polynomials {fk (X)} k∈N and show that, for a fixed k ∈ N and arbitrary X ∈ N, the period length of the simple continued fraction expansion of √fk (X) is constant. Furthermore, we show that the period lengths of √fk (X) go to infinity with k. For each member of the families involved, we show how to determine, in an easy fashion, the fundamental unit of the underlying quadratic field. We also demonstrate how the simple continued...
A. Bazylewicz (1976)
Acta Arithmetica
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D. Przeworska-Rolewicz (1969)
Studia Mathematica
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Marshall Hall (1971)
Acta Arithmetica
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Liang-Cheng Zhang (1997)
Acta Arithmetica
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H. Fiedler, W. Jurkat, O. Körner (1977)
Acta Arithmetica
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Mahta Khosravi, John A. Toth (2005)
Annales de l'institut Fourier
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Let be the error term in Weyl’s law for a 3-dimensional Riemannian Heisenberg manifold. We prove that , where is a specific nonzero constant and is an arbitrary small positive number. This is consistent with the conjecture of Petridis and Toth stating that .The idea of the proof is to use the Poisson summation formula to write the error term in a form which can be estimated by the method of the stationary phase. The similar result will be also proven in the -dimensional case. ...