Integral points in arithmetic progression on \(y^2= x(x^2-n^2)\)
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.
scientific article; zbMATH DE number 1423155
scientific article