The least inert prime in a real quadratic field
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Abstract:
In this paper, we prove that for any positive fundamental discriminant $D > 1596$, there is always at least one prime $p \leq D^{0.45}$ such that the Kronecker symbol $(D/p) = -1$. This improves a result of Granville, Mollin and Williams, where they showed that the least inert prime $p$ in a real quadratic field of discriminant $D > 3705$ is at most $\sqrt {D}/2$. We use a “smoothed” version of the Pólya–Vinogradov inequality, which is very useful for numerically explicit estimates.References
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Additional Information
- Enrique Treviño
- Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
- Address at time of publication: Department of Mathematics and Statistics, Swarthmore College, Swarthmore, Pennsylvania 19081
- ORCID: 0000-0002-7041-9814
- Email: etrevin1@swarthmore.edu
- Received by editor(s): May 10, 2011
- Received by editor(s) in revised form: June 8, 2011
- Published electronically: February 10, 2012
- Additional Notes: This paper is essentially Chapter 3 of the author’s Ph.D. Dissertation [16].
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 81 (2012), 1777-1797
- MSC (2010): Primary 11L40, 11Y40, 11R11
- DOI: https://doi.org/10.1090/S0025-5718-2012-02579-8
- MathSciNet review: 2904602