In this paper, we discuss issues related to the efficient implementation of Shanks' NUCOMP algorithm for computing the reduced composite of two binary quadratic forms. In particular, we describe how efficient versions of NUCOMP can be implemented for computations in imaginary quadratic number and function fields, as well as computations in the… (More)
Let p be a prime congruent to 1 modulo 4, and let t, u be rational integers such that (t + u √ p)/2 is the fundamental unit of the real quadratic field Q(√ p). The Ankeny-Artin-Chowla conjecture (AAC conjecture) asserts that p will not divide u. This is equivalent to the assertion that p will not divide B (p−1)/2 , where Bn denotes the nth Bernoulli number.… (More)
We note that the continued fraction expansion of a lacunary formal power series is a folded continued fraction with monomial partial quotients, and with the property that its convergents have denominators that are the sums of distinct monomials, that is, they are polynomials with coefficients 0, 1, and −1 only. Our results generalise, simplify and refine… (More)
We construct all families of quartic polynomials over Q whose square root has a periodic continued fraction expansion, and detail those expansions. In particular we prove that, contrary to expectation, the cases of period length nine and eleven do not occur. We conclude by providing a list of examples of pseudo-elliptic integrals involving square roots of… (More)
There is a class of quadratic number fields for which it is possible to find an explicit continued fraction expansion of a generator and hence an explicit formula for the fundamental unit. One there-with displays a family of quadratic fields with relatively large regulator. The formula for the fundamental unit seems far simpler than the continued fraction… (More)
Our investigations in the 1980s of Thue's method yielded determinants which we were only able to analyse successfully in part. We explain the context of our work, recount our experiences, mention our conjectures, and allude to a number of open questions.
We obtain new algorithms for testing whether a given by a black box multivariate polynomial over p-adic ÿelds given by a black box is identical to zero. We also remark on the zero testing of polynomials in residue rings. Our results complement a known results on the zero testing of polynomials over the integers, the rationals, and over ÿnite ÿelds.
We study certain two-dimensional
This note is a detailed explanation of Shanks–Atkin NUCOMP— composition and reduction carried out " simultaneously " —for all quadratic fields, that is, including real quadratic fields. That explanation incidentally deals with various " exercises " left for confirmation by the reader in standard texts. Extensive testing in both the numerical and function… (More)