• Corpus ID: 244896160

The average number of integral points on the congruent number curves

  title={The average number of integral points on the congruent number curves},
  author={Stephanie Chan},
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). 

Integral points on cubic twists of Mordell curves

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



Rational and Integral Points on Quadratic Twists of a Given Hyperelliptic Curve

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

Reduction of Binary Cubic and Quartic Forms

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

The second moment of the number of integral points on elliptic curves is bounded

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

A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan curves.

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

On the equivalence of binary quartics

Integral points on the congruent number curve

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

Elliptic Curves: Diophantine Analysis

I. General Algebraic Theory.- I. Elliptic Functions.- II. The Division Equation.- III. p-Adic Addition.- IV. Heights.- V. Kummer Theory.- V1. Integral Points.- II. Approximation of Logarithms.- VII.

The Probabilistic Method

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.