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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Corrigenda and addition to ``Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than $100\,000\,000\,000$''


Authors: A. J. van der Poorten, H. J. J. te Riele and H. C. Williams
Journal: Math. Comp. 72 (2003), 521-523
MSC (2000): Primary 11A55, 11J70, 11Y40, 11Y65, 11R11
Published electronically: October 16, 2002
Original Article: Math. Comp. 70 (2001), 1311-1328.
MathSciNet review: 1933835
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Abstract | References | Similar Articles | Additional Information

Abstract: 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 $\equiv5\bmod{8}$, the AAC conjecture has been verified using a different multiple of the regulator of the quadratic field $\mathbb{Q} (\sqrt{p})$ than was meant. However, since any multiple of this regulator is suitable for this purpose, provided that it is smaller than $8p$, the main result that the AAC conjecture is true for all the primes $\equiv1\bmod{4}$ which are $<10^{11}$, remains valid.

As an addition, we have verified the AAC conjecture for all the primes $\equiv1\bmod{4}$ between $10^{11}$ and $2\times10^{11}$, with the corrected program.


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  • 1. A. J. van der Poorten, H. J. J. te Riele and H. C. Williams, Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100000000000, Math. Comp, 70 (2001), 1311-1328. MR 2001j:11125

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Additional Information

A. J. van der Poorten
Affiliation: Centre for Number Theory Research, Macquarie University, Sydney, New South Wales 2109, Australia
Email: alf@math.mq.edu.au

H. J. J. te Riele
Affiliation: CWI, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands
Email: herman@cwi.nl

H. C. Williams
Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4
Email: williams@math.ucalgary.ca

DOI: http://dx.doi.org/10.1090/S0025-5718-02-01527-2
PII: S 0025-5718(02)01527-2
Keywords: Periodic continued fractions, function field
Received by editor(s): June 19, 2002
Published electronically: October 16, 2002
Article copyright: © Copyright 2002 American Mathematical Society