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 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 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)
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.
An error in the program for verifying the Ankeny-Artin-Chowla (AAC) conjecture is reported. As a result, in the case of primes p which are ≡ 5 mod 8, the AAC conjecture has been verified using a different multiple of the regulator of the quadratic field Q(√ p) than was meant. However, since any multiple of this regulator is suitable for this purpose,… (More)
Introduction The importance of recurrence sequences hardly needs to be explained. Their study is plainly of intrinsic interest and has been a central part of number theory for many years. Moreover, these sequences appear almost everywhere in mathematics and computer science. For example, the theory of power series representing rational functions ,… (More)
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)