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- Title
On pseudopoints of algebraic curves.
- Authors
Farashahi, Reza R.; Shparlinski, Igor E.
- Abstract
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 $${f \in \mathbb{Z}[U, V]}$$) 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.
- Publication
Archiv der Mathematik, 2010, Vol 95, Issue 6, p529
- ISSN
0003-889X
- Publication type
Article
- DOI
10.1007/s00013-010-0200-7