On pseudopoints of algebraic curves

Reza R. Farashahi, Igor E. Shparlinski

Research output: Contribution to journalArticlepeer-review


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 f∈ Z[U,V]) 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.

Original languageEnglish
Pages (from-to)529-537
Number of pages9
JournalArchiv der Mathematik
Issue number6
Publication statusPublished - Dec 2010


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