Alfred J. van der Poorten

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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 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)
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.
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)