Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000

A. J. Van Der Poorten; H. J J Te Riele; H. C. Williams

doi:10.1090/S0025-5718-00-01234-5

Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000

A. J. Van Der Poorten^*, H. J J Te Riele, H. C. Williams

^*Corresponding author for this work

Macquarie University Administration Unit

Research output: Contribution to journal › Article › peer-review

28 Citations (Scopus)

Abstract

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 B_n denotes the nth 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 < 10⁹. 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 10¹¹.

Original language	English
Pages (from-to)	1311-1328
Number of pages	18
Journal	Mathematics of Computation
Volume	70
Issue number	235
DOIs	https://doi.org/10.1090/S0025-5718-00-01234-5
Publication status	Published - Jul 2001

Bibliographical note

Corrigenda and addition can be found in Mathematics of Computation, 72(241), pp. 521-523, 2003.
http://doi.org/10.1090/S0025-5718-02-01527-2

Keywords

Function field
Periodic continued fraction

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10.1090/S0025-5718-00-01234-5

Cite this

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title = "Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000",

abstract = "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. 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 10330 000. In 1988 the AAC conjecture was verified by computer for all p < 109. 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 1011.",

keywords = "Function field, Periodic continued fraction",

author = "{Van Der Poorten}, {A. J.} and {Te Riele}, {H. J J} and Williams, {H. C.}",

note = "Corrigenda and addition can be found in Mathematics of Computation, 72(241), pp. 521-523, 2003. http://doi.org/10.1090/S0025-5718-02-01527-2",

year = "2001",

month = jul,

doi = "10.1090/S0025-5718-00-01234-5",

language = "English",

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AU - Te Riele, H. J J

AU - Williams, H. C.

N1 - Corrigenda and addition can be found in Mathematics of Computation, 72(241), pp. 521-523, 2003. http://doi.org/10.1090/S0025-5718-02-01527-2

PY - 2001/7

Y1 - 2001/7

N2 - 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. 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 10330 000. In 1988 the AAC conjecture was verified by computer for all p < 109. 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 1011.

AB - 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. 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 10330 000. In 1988 the AAC conjecture was verified by computer for all p < 109. 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 1011.

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KW - Periodic continued fraction

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ER -

Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000

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