Open Access
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


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.


Download Citation

Wolfgang Knapp. Markus Köcher. Peter Schmid. "Quadratic residues and class numbers." Funct. Approx. Comment. Math. 47 (2) 173 - 182, December 2012.


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

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

Keywords: class numbers , primes , Quadratic forms , quadratic residues , Siegel--Tatuzawa

Rights: Copyright © 2012 Adam Mickiewicz University

Vol.47 • No. 2 • December 2012
Back to Top