On pseudopoints of algebraic curves

Following Kraitchik and Lehmer, we say that a positive integer n ≡ 1 (mod 8) is an x-pseudosquare if it is a quadratic residue for each odd prime p ≤ x, yet it is not a square. We extend this definition to algebraic curves and say that n is an x-pseudopoint of a curve defined by f(U, V) = 0 (where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f \in \mathbb{Z}[U, V]}$$\end{document}) if for all sufficiently large primes p ≤ x the congruence f(n, m) ≡ 0 (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.