Quadratic residues and class numbers

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.