2010/05/26 by Reza R. Farashahi, Farashahi, Reza R., Igor E. Shparlinski +1
Mathematics · #11G20 #11T23 #11Y40 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:11G20 #msc:11T23 #msc:11Y40
paper · pdf · doi:10.48550/arxiv.1005.4775
arxiv created 2010/05/26 · arxiv updated 2010/10/22
Following Kraitchik and Lehmer, we say that a positive integer n≡1\pmod 8 is an x-pseudosquare if it is a quadratic residue for each odd prime p≤ 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 ∈ \Z[U,V]) if for all sufficiently large primes p ≤ x the congruence f(n,m)≡ 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.