For uniformly chosen random α ∈ [0, 1], it is known the probability the nth digit of the continued-fraction expansion, [α]n converges to the Gauss-Kuzmin distribution P([α]n = k) ≈ log2(1 + 1/k(k + 2)) as n → ∞. In this paper, we show the continued fraction digits of √ d, which are eventually periodic, also converge to the Gauss-Kuzmin distribution as d → ∞ with bounded class number, h(d). The ...

It is well known that the regular continued fraction expansion of a quadratic irrational is symmetric about its centre; we refer to this symmetry as horizontal. However, an additional vertical symmetry is exhibited by the continued fraction expansions arising from a family of quadratics known as Schinzel sleepers. This paper provides a method for generating every Schinzel sleeper and investigat...

In 1986, Mills and Robbins observed by computer the continued fraction expansion of certain algebraic power series over a finite field. Incidentally, they came across a particular equation of degree four in characteristic p = 13. This equation has an analogue for all primes p ≥ 5. There are two patterns for the continued fraction of the solution of this equation, according to the residue of p m...

We detail the continued fraction expansion of the square root of the general monic quartic polynomial. We note that each line of the expansion corresponds to addition of the divisor at infinity, and interpret the data yielded by the general expansion. The paper includes a detailed ’reminder exposition’ on continued fractions of quadratic irrationals in function fields. A delightful ‘essay’ [16]...

In this paper, we present a new algorithm for computing the reduced sum of two divisors of an arbitrary hyperelliptic curve. Our formulas and algorithms are generalizations of Shanks’s NUCOMP algorithm, which was suggested earlier for composing and reducing positive definite binary quadratic forms. Our formulation of NUCOMP is derived by approximating the irrational continued fraction expansion...

It is, in general, very hard to predict the features of the continued fraction expansion of a given positive real number. If the number in question is of the form √ d, where d is a positive integer which is not a square, then its continued fraction expansion is of the form [a0, {a1, . . . , ar−1, 2a0}], where we use {. . . } to emphasize the period of the expansion. It is known that a1, . . . ,...

We provide here a complete average–case analysis of the binary continued fraction representation of a random rational whose numerator and denominator are odd and less than N . We analyse the three main parameters of the binary continued fraction expansion, namely the height, the number of steps of the binary Euclidean algorithm, and finally the sum of the exponents of powers of 2 contained in t...

The lattice reduction algorithm of Gauss is shown to have an average case complexity which is asymptotic to a constant. Introduction. The “reduction” algorithm of Gauss plays an important r6le in several areas of computational number theory, principally in matters related to the reduction of integer lattice bases. It is also intimately connected with extensions to complex numbers of the Euclide...

An S-adic expansion of an infinite word is a way of writing it as the limit of an infinite product of substitutions (i.e., morphisms of a free monoid). Such a description is related to continued fraction expansions of numbers and vectors. Indeed, with a word is naturally associated, whenever it exists, the vector of frequencies of its letters. A fundamental example of this relation is between S...