TY - JOUR

T1 - On pseudopoints of algebraic curves

AU - Farashahi, Reza R.

AU - Shparlinski, Igor E.

PY - 2010/12

Y1 - 2010/12

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=78649984633&partnerID=8YFLogxK

U2 - 10.1007/s00013-010-0200-7

DO - 10.1007/s00013-010-0200-7

M3 - Article

AN - SCOPUS:78649984633

VL - 95

SP - 529

EP - 537

JO - Archiv der Mathematik

JF - Archiv der Mathematik

SN - 0003-889X

IS - 6

ER -