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

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

2000-05-14

arxiv.org/abs/math/0005139v1

Mathematics

Number Theory

31 pages, including 2 numerical tables; invited talk at ANTS-IV (4th Algorithmic Number Theory Symposium, Leiden 7/2000). Simi

Scientific paper

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|>N, using only O(log N) space. Since the number of solutions should be asymptotically proportional to M log N (as long as M

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