Geodesic
Polynomials of Pellian Type and Continued Fractions
Serdica Mathematical Journal, Tome 27 (2001) no. 4, pp. 317-342

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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 fraction ex- pansion of √fk (X) is related to that of √C, where √fk (X) = ak*X^2 +bk*X + C. This continues work in [1]–[4].
Keywords: Continued Fractions, Pell’s Equation, Period Length 
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     author = {Mollin, R.},
     title = {Polynomials of {Pellian} {Type} and {Continued} {Fractions}},
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Mollin, R. Polynomials of Pellian Type and Continued Fractions. Serdica Mathematical Journal, Tome 27 (2001) no. 4, pp. 317-342. http://geodesic.mathdoc.fr/item/SMJ2_2001_27_4_a4/