Algorithmic Number Theory

Volume 1838 of the series Lecture Notes in Computer Science pp 33-63

Rational Points Near Curves and Small Nonzero | x 3y 2| via Lattice Reduction

  • Noam D. ElkiesAffiliated withDepartment of Mathematics, Harvard University

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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 < xy < z < N in heuristic time ≪ (log O(1) N ) M provided MN, 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 as low as possible. Moreover the algorithm readily parallelizes. It not only yields new numerical examples but leads to theoretical results, difficult open questions, and natural generalizations. We also adapt our algorithm to investigate Hall’s conjecture: we find all integer solutions of 0 < |x 3y 2| ≪x 1/2 with x < X in time O(X 1/2log O(1) X). By implementing this algorithm with X = 1018 we shattered the previous record for x 1/2/|x 3y 2|. The O(X 1/2log O(1) X) bound is rigorous; its proof also yields new estimates on the distribution mod 1 of (cx)3/2 for any positive rational c.