Rational points near curves and small nonzero |x^3-y^2| via lattice reduction

Abstract

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 << L(N) M [where L(X):= (log X)^O(1)] 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 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^3-y^2|<<x^(1/2) with x<X in time O(X^(1/2) L(X)). By implementing this algorithm with X=10^18 we shattered the previous record for x^(1/2)/|x^3-y^2|. The O(X^{1/2} L(X)) bound is rigorous; its proof also yields new estimates on the distribution mod 1 of sqrt(cx^3) for any positive rational c.