On the size of the fundamental solution of the Pell equation (Q305146)

Given a Pell equation NEWLINE\[NEWLINEt^2-Du^2=1NEWLINE\]NEWLINE it is well-known that all solutions \(\eta_D=t+u\sqrt{D}\) to this Pell equation can be written in the form \(\eta_D=\pm \varepsilon_D^n\), where \(\epsilon_D\) is the so-called fundamental solution. One of the most interesting question concerning Pell equations is how large these fundamental solutions can get. However, in this paper the problem is investigated how many positive non-square integers \(D\) exist such that \(\varepsilon_D\) stays small. More precisely let NEWLINE\[NEWLINES(x,\alpha):=\left\{ (\eta_D,D): 2\leq D\leq x,\;D \text{ is non-square, } \varepsilon_D\leq \eta_D \leq D^{1/2+\alpha}\right\}NEWLINE\]NEWLINE and NEWLINE\[NEWLINES^f(x,\alpha):=\left\{ (\varepsilon_D,D): 2\leq D\leq x,\;D \text{ is non-square, } \varepsilon_D\leq D^{1/2+\alpha}\right\}.NEWLINE\]NEWLINE \textit{C. Hooley} [J. Reine Angew. Math. 353, 98--131 (1984; Zbl 0539.10019)] obtained for \(0<\alpha< 1/2\) that NEWLINE\[NEWLINES(x,\alpha)\sim S^f(x,\alpha)\sim \frac{4\alpha^2}{\pi^2}x^{1/2}\log^2 x.NEWLINE\]NEWLINE The main result in the paper are lower bounds uniformly in \(\tfrac 12\leq \alpha \leq 1\) for \(S(x,\alpha)\) and \(S^f(x,\alpha)\) and therefore extending Hooley's results.NEWLINENEWLINENEWLINEAs the author points out finding integers \(D\) such that \(\eta_D\) is small is linked with the study of the congruence \(t^2-1\equiv 0\bmod u^2\). Explicit solutions can be described by the factorization of \(u=u_1u_2\) with \(\gcd(u_1,u_2)=1\). Using Fourier analysis the author is led to study a tridimensional exponential sum.NEWLINENEWLINEAssuming a conjecture on short exponential sums even stronger results are obtained by the author.