Fundamental solutions to Pell equation with prescribed size
Let \(\varepsilon_D=T+U\sqrt D>1\) be the fundamental solution to the Pell equation \(T^2-DU^2=1\) and let \[ S^f(x;\alpha_1,\alpha_2):=\sharp \{D; \; D\leq x, D\neq \square, D^{1/2+\alpha_1}\leq \varepsilon_D \leq D^{1/2+\alpha_2} \}. \] \textit{C. Hooley} [J. Reine Angew. Math. 353, 98--131 (1984; Zbl 0539.10019)] conjectured that for every pair of real numbers \((\alpha_1,\alpha_2)\) such that \(0\leq \alpha_1<\alpha_2\) there exists \(c(\alpha_1,\alpha_2)>0\) and \(X(\alpha_1,\alpha_2)\) such that, \[ S^f(x;\alpha_1,\alpha_2)\geq c(\alpha_1,\alpha_2)x^{1/2} (\log x)^2 \] provided \(x\geq X(\alpha_1,\alpha_2)\). The main result of the paper under review is that the conjecture is true if \(\alpha_1<3/2\). The main ingredient in the proof is to transform the problem into a suitable character sum and use the famous Polya-Vinogradov inequality to estimate this sum.
scientific article; zbMATH DE number 6345303