• Corpus ID: 244896160

# The average number of integral points on the congruent number curves

@inproceedings{Chan2021TheAN,
title={The average number of integral points on the congruent number curves},
author={Stephanie Chan},
year={2021}
}
We show that the total number of non-torsion integral points on the curves ED : y = x − Dx, where D ranges over positive squarefree integers less than N , is ≪ N(logN).
1 Citations
Fix a non-square integer $k\neq 0$. We show that the number of curves $E_B:y^2=x^3+kB^2$ containing an integral point, where $B$ ranges over positive integers less than $N$, is bounded by $O_k(N(\log ## References SHOWING 1-10 OF 20 REFERENCES We show that the abc-conjecture implies that few quadratic twists of a given hyperelliptic curve have any non-trivial rational or integral points; and indicate how these considerations dovetail with A reduction theory is developed for binary forms (homogeneous polynomials) of degrees three and four with integer coefficients. The resulting coefficient bounds simplify and improve on those in the • Mathematics • 2018 In this paper, we show that the second moment of the number of integral points on elliptic curves over$\mathbb{Q}$is bounded. In particular, we prove that, for any$0 < s < \log_2 5 = 2.3219
The two fundamental finiteness theorems in the arithmetic theory of elliptic curves are the Mordell-Weil theorem, which says that the group of rational points is finitely generated, and Siegel's
We study integral points on the quadratic twists $\mathcal{E}_D:y^2=x^3-D^2x$ of the congruent number curve. We give upper bounds on the number of integral points in each coset of
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• Computer Science
SODA
• 1992
A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.