Rational Points Near Curves and Small Nonzero |x3-y2| via Lattice Reduction

  title={Rational Points Near Curves and Small Nonzero |x3-y2| via Lattice Reduction},
  author={Noam D. Elkies},
  • N. Elkies
  • Published in ANTS 14 May 2000
  • Mathematics
We give a new algorithm using linear approximation and lattice reduction to efficiently calculate all rational points of small height near a given plane curve C. For instance, when C is the Fermat cubic, we find all integer solutions of | x 3 + y 3 −z 3| < M with 0 < x ≤y < z < N in heuristic time ≪ (log O(1) N ) M provided M ≫N, using only O(log N) space. Since the number of solutions should be asymptotically proportional to M log N (as long as M < N 3), the computational costs are essentially… 
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