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

TL;DR

This paper introduces a new lattice reduction algorithm for efficiently finding rational points near curves, with applications to Fermat's Last Theorem and Hall's conjecture, achieving near-optimal computational complexity.

Contribution

The authors develop a novel algorithm combining linear approximation and lattice reduction to find small solutions near curves, improving computational efficiency and enabling new theoretical insights.

Findings

Successfully finds all solutions of |x^3 + y^3 - z^3| < M for given bounds

Breaks previous records in solving |x^3 - y^2| problems up to 10^18

Provides new estimates on the distribution mod 1 of sqrt(cx^3)

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.