  Following Kraitchik and Lehmer, we say that a positive integer $n\equiv1\pmod
8$ 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 $f \in \Z[U,V]$)
if for all sufficiently large primes $p \le x$ the congruence $f(n,m)\equiv 0
\pmod p$ 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.
