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

```@inproceedings{Elkies2000RationalPN,
title={Rational Points Near Curves and Small Nonzero |x3-y2| via Lattice Reduction},
author={Noam D. Elkies},
booktitle={ANTS},
year={2000}
}```
• 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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## References

SHOWING 1-10 OF 28 REFERENCES
On searching for solutions of the Diophantine equation x3 + y3 + z3 = n
• Mathematics, Computer Science
Math. Comput.
• 1997
A new search algorithm to solve the equation x 3 + y 3 + z 3 = n for a fixed value of n > 0.5 is proposed, using several efficient number-theoretic sieves and much faster on average than previous straightforward algorithms.
Geometric postulation of a smooth function and the number of rational points
This paper is devoted to giving refinements and extensions of some of the results of Bombieri and the author [1] obtaining upper bounds for the number of integral lattice points on the graphs of
On Mordell's Equation
• Mathematics
Compositio Mathematica
• 1998
In an earlier paper we developed an algorithm for computing all integral points on elliptic curves over the rationals Q. Here we illustrate our method by applying it to Mordell's Equation y2=x3+k for
The number of integral points on arcs and ovals
• Mathematics
• 1989
integral lattice points, and that the exponent and constant are best possible. However, Swinnerton–Dyer [10] showed that the preceding result can be substantially improved if we start with a fixed,
Heegner point computations
• N. Elkies
• Mathematics, Computer Science
ANTS
• 1994
We discuss the computational application of Heegner points to the study of elliptic curves over Q, concentrating on the curves E d : Dy2 = x3 − x arising in the “congruent number” problem. We begin
The density of zeros of forms for which weak approximation fails
The weak approximation principal fails for the forms x + y + z = kw, when k = 2 or 3. The question therefore arises as to what asymptotic density one should predict for the rational zeros of these
Algorithms for Modular Elliptic Curves
This book presents a thorough treatment of many algorithms concerning the arithmetic of elliptic curves with remarks on computer implementation and an extensive set of tables giving the results of the author's implementations of the algorithms.
Shimura Curve Computations
Some methods for computing equations for certain Shimura curves, natural maps between them, and special points on them are given, and a list of open questions that may point the way to further computational investigation of these curves are illustrated.
Elliptic and modular curves over finite fields and related computational issues
The problem of calculating the trace of an elliptic curve over a finite field has attracted considerable interest in recent years. There are many good reasons for this. The question is intrinsically
Old and new conjectured diophantine inequalities
The original meaning of diophantine problems is to find all solutions of equations in integers or rational numbers, and to give a bound for these solutions. One may expand the domain of coefficients