Displaying similar documents to “On some problems of the arithmetical theory of continued fractions”

Cavity method in the spherical SK model

Dmitry Panchenko (2009)

Annales de l'I.H.P. Probabilités et statistiques

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We develop a cavity method for the spherical Sherrington–Kirkpatrick model at high temperature and small external field. As one application we compute the limit of the covariance matrix for fluctuations of the overlap and magnetization.

Polynomials of Pellian Type and Continued Fractions

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...

Halfway to a solution of X 2 - D Y 2 = - 3

R. A. Mollin, A. J. Van der Poorten, H. C. Williams (1994)

Journal de théorie des nombres de Bordeaux

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It is well known that the continued fraction expansion of D readily displays the midpoint of the principal cycle of ideals, that is, the point halfway to a solution of x 2 - D y 2 = ± 1 . Here we notice that, analogously, the point halfway to a solution of x 2 - D y 2 = - 3 can be recognised. We explain what is going on.