Some More Long Continued Fractions, I
نویسندگان
چکیده
In this paper we show how to construct several infinite families of polynomials D(x̄, k), such that p D(x̄, k) has a regular continued fraction expansion with arbitrarily long period, the length of this period being controlled by the positive integer parameter k. We also describe how to quickly compute the fundamental units in the corresponding real quadratic fields.
منابع مشابه
Some New Families of Tasoevian- and Hurwitzian Continued Fractions
We derive closed-form expressions for several new classes of Hurwitzianand Tasoevian continued fractions, including [0; p− 1, 1, u(a + 2nb)− 1, p− 1, 1, v(a + (2n + 1)b)− 1 ]n=0, [0; c + dmn]n=1 and [0; eun, fvn] ∞ n=1. One of the constructions used to produce some of these continued fractions can be iterated to produce both Hurwitzianand Tasoevian continued fractions of arbitrary long quasi-pe...
متن کاملSome Facts About Continued Fractions That Should Be Better Known
In this report I will give proofs of some simple theorems concerning continued fractions that are known to the cognoscenti, but for which proofs in the literature seem to be missing, incomplete, or hard to locate. In particular, I will give two proofs of the following “folk theorem”: if θ is an irrational number whose continued fraction has bounded partial quotients, then any non-trivial linear...
متن کاملDistribution of zinc and copper chemical forms and their relationship with some physico-chemical properties and clay minerals in some calcareous soils
Determination and recognition of relative distribution of chemical forms of each element and their relationship with physical, chemical and soil clay minerals can help researchers to manage soil fertility better. This research attempted to recognize chemical fractions of zinc (Zn) and copper (Cu) in some surface and subsurface soil samples of Kohgiluyeh and Boyer-Ahmad province and their relati...
متن کاملDimension of Some Non - normal Continued Fraction Sets
We consider certain sets of non-normal continued fractions for which the asymptotic frequencies of digit strings oscillate in one or other ways. The Hausdorff dimensions of these sets are shown to be the same value 1/2 as long as they are non-empty. An outstanding example of such sets is the set of “extremely non-normal continued fractions” which was previously conjectured to be of Hausdorff di...
متن کاملA ug 2 00 1 CONTINUED FRACTIONS , MODULAR SYMBOLS , AND NON – COMMUTATIVE GEOMETRY
Abstract. Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss–Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive convergents. This result has an application to the Mixmaster Universe model in general relativity. We then study some averages involving modular s...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2006