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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. 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
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Abstract: Let $p$ be a prime congruent to 1 modulo 4, and let $t, u$ be rational integers such that $(t+u\sqrt{p}\,)/2$ is the fundamental unit of the real quadratic field $\mathbb{Q}(\sqrt{p}\,)$. 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 $B_{n}$ denotes the $n$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 $t, u$; for example, when $p = 40\,094\,470\,441$, then both $t$ and $u$ exceed $10^{330\,000}$. In 1988 the AAC conjecture was verified by computer for all $p < 10^{9}$. 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 $p$ up to $10^{11}$.

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Additional Information

A. J. van der Poorten
Affiliation: Centre for Number Theory Research, Macquarie University, Sydney, NSW 2109, \penalty10000 Australia

H. J. J. te Riele
Affiliation: CWI, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands

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

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