On Pseudopoints of Algebraic Curves

Abstract

Following Kraitchik and Lehmer, we say that a positive integer $n\equiv 1(\mathrm{mod}8)$ is an $x$-pseudosquare if it is a quadratic residue for each odd prime $p\le x$, yet is not a square. We extend this defintion to algebraic curves and say that $n$ is an $x$-pseudopoint of a curve $f(u,v)=0$ (where $f\in \mathbb{Z}[U,V]$) if for all sufficiently large primes $p\le x$ the congruence $f(n,m)\equiv 0(\mathrm{mod}p)$ is satisfied for some $m$. We use the Bombieri bound of exponential sums along a curve to estimate the smallest $x$-pseudopoint, which shows the limitations of the modular approach to searching for points on curves.