The Class-Number of Real Quadratic Number Fields

Ankeny, N. C.; Artin, Emil; Chowla, S.

doi:10.2307/1969656

Cited by 73 publications

(54 citation statements)

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“…In 1952 Ankeny, Artin and Chowla [2] asked whether p u always and noted that p u for p < 2000 (p ≡ 5 (mod 8)). This question was written in the form of a conjecture by Mordell [15], and has since become known as the Ankeny-Artin-Chowla conjecture (AAC conjecture).…”

Section: Introductionmentioning

confidence: 99%

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Computer verification of the Ankeny--Artin--Chowla Conjecture for all primes less than $100000000000$

2000

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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 √ 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 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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Section: Introductionmentioning

confidence: 99%

“…It arose ultimately from expressions which were derived in [2] for the value of hu/t modulo p, where h is the class number of Q( √ p ). One of these results is hu/t ≡ B (p−1)/2 (mod p), (1.1) .…”

Section: Introductionmentioning

confidence: 99%

Computer verification of the Ankeny--Artin--Chowla Conjecture for all primes less than $100000000000$

2000

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“…Indeed, if ε t = 2 Z for some 2 of A, then iV fc/Q(Λ/ τ ) ε ι = 2', also el ι~1)/2 = z\ where z t = N lc/Q{^T) z. Hence ^ is a unit of Q(V Z) and we have ε t -ε\jε\~ι = (εjz 2 ) 1 . Since εj is the fundamental unit of Q(V Z), there is a rational integer c such that ε L !z 2 = ±εj.…”

Section: Lei Z = 1 (Mod 4) αRccz Eι = (T + Uvt)/2 > 1 Be the Fundamenmentioning

confidence: 91%

On the unramified extensions of the prime cyclotomic number field and its quadratic extensions

Nakagoshi

1989

Nagoya Mathematical Journal

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It is interesting to know what kinds of primes are the factors of the class number of an algebraic number field, and especially to find ones being prime to the degree. About this matter it is desirable to construct the unramified Abelian extensions plainly. In this paper we shall show some of them for the prime cyclotomic number field and its quadratic extensions using the units of subfields.Let I be an odd prime and ζ be a primitive /-th root of unity. Let k = Q(ζ) be the /-th cyclotomic number field over the field Q of rationals. If / is irregular, then there is an even integer r with 2 1 of Q(

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“…according to (2) and (3). Therefore we can rewrite the right-hand side of the recurrence relation as…”

Section: ä ññ 21ºmentioning

confidence: 99%

“…The Ankeny-Artin-Chowla congruence [1,2] The Ankeny-Artin-Chowla Conjecture (which remains open to this day) asserts that p does not divide U . This is equivalent to the assertion that p does not divide B p−1…”

Section: Introduction and Notationmentioning

confidence: 99%