Integral points in arithmetic progression on \(y^2= x(x^2-n^2)\)
type: Item
instance of: scholarly article
title: Integral points in arithmetic progression on \(y^2= x(x^2-n^2)\)
zbMATH Open document ID: 1009.11035
author: Joseph H. Silverman
author: Andrew Bremner
author: Nikos Tzanakis
published in: Journal of Number Theory
publication date: 2003-04-29T00:00:00Z
review text: Let \(n\geq 1\) and consider the elliptic curve \(E: y^2=x(x^2-n^2)\). The authors consider the problem of finding three integral points \(P_i\) on the curve such that their \(x\)-coordinates \(x_i=x(P_i)\) form an arithmetic progression \(x_1<x_2<x_3\). The main goal of the paper is to show that, if \(n\geq 1\) is square-free, then a subgroup \(\Gamma\) of \(E(Q)\) of rank 1 contains no non-trivial integral arithmetic progressions. The proof of this statement requires height computations to deal with the cases \(n\geq 72\) and explicit computations for \(1\leq n<72\). The authors describe those cases when the arithmetic progression contains one of the torsion points \(T_1=(-n,0), T_2=(0,0), T_3=(n,0)\) and two non-torsion points.
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 1423155
zbMATH Keywords: elliptic curves
zbMATH Keywords: arithmetic progressions
reviewed by: István Gaál
MaRDI profile type: Publication
full work available at URL: jnth.1999.2430
OpenAlex ID: W2077954256
cites work: On squares of squares
cites work: On the Equation Y 2 = X(X 2 + p)
cites work: On sums of consecutive squares
cites work: On the Conjecture of Birch and Swinnerton-Dyer for an Elliptic Curve of Rank 3
cites work: The Diophantine equation x⁴ - Dy² = 1, II
cites work: Estimate of the remainder term in Tate's formula
cites work: Computing integral points on elliptic curves
cites work: Q4378071
cites work: The canonical height and integral points on elliptic curves
cites work: Q4168661
cites work: Magic Squares of Squares
cites work: Q3710637
cites work: Q4317891
cites work: Lower bound for the canonical height on elliptic curves
cites work: Lower bounds for height functions
cites work: Computing Heights on Elliptic Curves
cites work: The Difference Between the Weil Height and the Canonical Height on Elliptic Curves
cites work: Infinite descent on elliptic curves
cites work: Q4855233
cites work: Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms
cites work: Algorithm for determining the type of a singular fiber in an elliptic pencil
cites work: Computations of the rank of elliptic curve \(y^ 2=x^ 3-n^ 2x\)
cites work: On the difference of the Weil height and the Neron-Tate height
DOI: 10.1006/JNTH.1999.2430
description: scientific article; zbMATH DE number 1423155