Abstract
We give a lower bound for the cardinality of the set of
Funding statement: This research was partially supported by Institut Universitaire de France. The author thanks this institution.
The author thanks the referees very warmly. The exposition in this paper greatly benefited from their comments, questions and suggestions.
References
[1] Bourgain J., A remark on solutions of the Pell equation, Int. Math. Res. Not. IMRN (2014), 10.1093/imrn/rnu023. 10.1093/imrn/rnu023Search in Google Scholar
[2] Cohen H., Sur la distribution asymptotique des groupes de classes, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), 245–247. Search in Google Scholar
[3] Cohen H. and Lenstra H. W., Heuristics on class groups of number fields, Number theory (Noordwijkerhout 1983)., Lecture Notes in Math. 1068, Springer-Verlag, Berlin (1984), 33–62. 10.1007/BFb0099440Search in Google Scholar
[4] Estermann T., On Kloosterman’s sum, Mathematika 8 (1961), 83–86. 10.1112/S0025579300002187Search in Google Scholar
[5] Fouvry E. and Jouve F., A positive density of fundamental discriminants with large regulator, Pacific J. Math. 262 (2013), no. 1, 81–107. 10.2140/pjm.2013.262.81Search in Google Scholar
[6] Fouvry E. and Jouve F., Size of regulators and consecutive square-free numbers, Math. Z. 273 (2013), no. 3–4, 869–882. 10.1007/s00209-012-1035-7Search in Google Scholar
[7]
Golubeva E. P.,
The class numbers of real quadratic fields of discriminant
[8] Golubeva E. P., On the Pellian equation, J. Math. Sci. (N. Y.) 122 (2004), 3600–3602. 10.1023/B:JOTH.0000035233.98166.18Search in Google Scholar
[9] Heath-Brown D. R., A mean value estimate for real character sums, Acta Arith. 72 (1995), 235–275. 10.4064/aa-72-3-235-275Search in Google Scholar
[10] Hooley C., On the greatest prime factor of a cubic polynomial, J. reine angew. Math. 303/304 (1978), 921–950. 10.1515/crll.1978.303-304.21Search in Google Scholar
[11] Hooley C., On the Pellian equation and the class number of indefinite binary quadratic forms, J. reine angew. Math. 353 (1984), 98–131. 10.1515/crll.1984.353.98Search in Google Scholar
[12] Hooley C., On Waring’s problem, Acta Math. 157 (1986), 49–97. 10.1007/BF02392591Search in Google Scholar
[13] Hua L. K., Introduction to number theory, Springer-Verlag, Berlin 1982. Search in Google Scholar
[14] Reuss T., Pairs of k-free numbers, consecutive square-full numbers, preprint 2014, http://arxiv.org/abs/1212.3150. Search in Google Scholar
[15] Sarnak P., Class numbers of indefinite binary quadratic forms. II, J. Number Theory 21 (1985), 333–346. 10.1016/0022-314X(85)90060-5Search in Google Scholar
[16] Sarnak P., Reciprocal geodesics, Analytic number theory, Clay Math. Proc. 7, American Mathematical Society, Providence (2007), 217–237. Search in Google Scholar
[17] Tenenbaum G., Introduction to analytic and probabilistic number theory, Cambridge Stud. Adv. Math. 46, Cambridge University Press, Cambridge 1995. Search in Google Scholar
[18] Weil A., Number theory: An approach through history, Birkhäuser-Verlag, Boston 1984. Search in Google Scholar
© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- On the size of the fundamental solution of the Pell equation
- Euler factors determine local Weil representations
- Projective compactifications and Einstein metrics
- Hilbert schemes and toric degenerations for low degree Fano threefolds
- Une base explicite de symboles modulaires sur les corps de fonctions
- Refined global Gan–Gross–Prasad conjecture for Bessel periods
- Bergman kernel along the Kähler–Ricci flow and Tian’s conjecture
- Bargmann–Fock extension from singular hypersurfaces
Articles in the same Issue
- Frontmatter
- On the size of the fundamental solution of the Pell equation
- Euler factors determine local Weil representations
- Projective compactifications and Einstein metrics
- Hilbert schemes and toric degenerations for low degree Fano threefolds
- Une base explicite de symboles modulaires sur les corps de fonctions
- Refined global Gan–Gross–Prasad conjecture for Bessel periods
- Bergman kernel along the Kähler–Ricci flow and Tian’s conjecture
- Bargmann–Fock extension from singular hypersurfaces