On the negative Pell equation

• 成果类型: Article
• 署名作者: Fouvry, Etienne; Klueners, Juergen
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2010.172.2035
• 发表日期: 2010
• 页码: 2035-2104
• 关键词: quadratic number bodies; 4 divisible invariants; absolute class group; FIELDS; heuristics; THEOREM; sums

摘要

We give asymptotic upper and lower bounds for the number of squarefree d (0 < d <= X) such that the equation x(2) - dy(2) = -1 is solvable. These estimates, as usual, can equivalently be interpreted in terms of real quadratic fields with a fundamental unit with norm -1 and give strong evidence in the direction of a conjecture due to P. Stevenhagen.