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December 2012 Quadratic residues and class numbers
Wolfgang Knapp, Markus Köcher, Peter Schmid
Funct. Approx. Comment. Math. 47(2): 173-182 (December 2012). DOI: 10.7169/facm/2012.47.2.4

Abstract

For an odd prime $p$ let $\rho_p$ be the least odd prime ($\ne p$) which is a~quadratic residue mod $p$. Using the theorems of Heegner--Baker--Stark and Siegel--Tatuzawa on the class number $h=h(-p)$ of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-p})$ it is shown that $\rho_p<\sqrt p$ unless $p\in \{3, 5, 7, 17, 19, 43, 67, 163\}$, possibly with one further exceptional (large) prime $p=p_u$ (satisfying $p=2^{h+2}-u^2$ with $h>100$ und $5\le u<2^{(h-5)/2}$). The exceptional prime does not exist if the Extended Riemann Hypothesis is true.

Citation

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Wolfgang Knapp. Markus Köcher. Peter Schmid. "Quadratic residues and class numbers." Funct. Approx. Comment. Math. 47 (2) 173 - 182, December 2012. https://doi.org/10.7169/facm/2012.47.2.4

Information

Published: December 2012
First available in Project Euclid: 20 December 2012

zbMATH: 1315.11005
MathSciNet: MR3051447
Digital Object Identifier: 10.7169/facm/2012.47.2.4

Subjects:
Primary: 11A15, 11E41
Secondary: 11A41, 11M20, 11R29

Rights: Copyright © 2012 Adam Mickiewicz University

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Vol.47 • No. 2 • December 2012
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