Computer verification of the Ankeny-Artin-Chowla Conjecture for all primes less than

Authors:
A. J. van der Poorten, H. J. J. te Riele and H. C. Williams

Journal:
Math. Comp. **70** (2001), 1311-1328

MSC (2000):
Primary 11A55, 11J70, 11Y40, 11Y65, 11R11

Published electronically:
March 15, 2000

Corrigendum:
Math. Comp. 72 (2002), 521-523.

MathSciNet review:
1709160

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a prime congruent to 1 modulo 4, and let be rational integers such that is the fundamental unit of the real quadratic field . The Ankeny-Artin-Chowla conjecture (AAC conjecture) asserts that will not divide . This is equivalent to the assertion that will not divide , where denotes the th Bernoulli number. Although first published in 1952, this conjecture still remains unproved today. Indeed, it appears to be most difficult to prove. Even testing the conjecture can be quite challenging because of the size of the numbers ; for example, when , then both and exceed . In 1988 the AAC conjecture was verified by computer for all . In this paper we describe a new technique for testing the AAC conjecture and we provide some results of a computer run of the method for all primes up to .

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Additional Information

**A. J. van der Poorten**

Affiliation:
Centre for Number Theory Research, Macquarie University, Sydney, NSW 2109, \penalty10000 Australia

Email:
alf@math.mq.edu.au

**H. J. J. te Riele**

Affiliation:
CWI, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands

Email:
Herman.te.Riele@cwi.nl

**H. C. Williams**

Affiliation:
Dept. of Computer Science, University of Manitoba, Winnipeg, Manitoba Canada R3T 2N2

Email:
williams@cs.umanitoba.ca

DOI:
http://dx.doi.org/10.1090/S0025-5718-00-01234-5

Keywords:
Periodic continued fraction,
function field

Received by editor(s):
March 22, 1999

Received by editor(s) in revised form:
July 6, 1999

Published electronically:
March 15, 2000

Additional Notes:
The first author was supported in part by a grant from the Australian Research Council.

The research of the third author was supported by NSERC Canada grant #A7649.

Article copyright:
© Copyright 2000
American Mathematical Society