On θ-Congruent Numbers, Rational Squares in Arithmetic Progressions, Concordant Forms and Elliptic Curves

Erich Selder,Karlheinz Spindler

doi:10.3390/math3010002

Abstract

The correspondence between right triangles with rational sides, triplets of rational squares in arithmetic succession and integral solutions of certain quadratic forms is well-known. We show how this correspondence can be extended to the generalized notions of rational θ-triangles, rational squares occurring in arithmetic progressions and concordant forms. In our approach we establish one-to-one mappings to rational points on certain elliptic curves and examine in detail the role of solutions of the θ-congruent number problem and the concordant form problem associated with nontrivial torsion points on the corresponding elliptic curves. This approach allows us to combine and extend some disjoint results obtained by a number of authors, to clarify some statements in the literature and to answer some hitherto open questions.

Highlights

The following definition dates back to Euler ([1]; see [2]).Definition 1
After factoring out the greatest common divisor of the coefficients, this leads us to consider quadratic forms X 2 − pkY 2 and X 2 + qkY 2 where k, p, q ∈ N with p and q coprime. Concordant forms in this form tie up nicely with rational squares occurring in arithmetic progressions
We note that arithmetic progressions of squares have been studied not just over the rationals, but over number fields. While in these approaches the goal was to find uninterrupted arithmetic progressions of squares in the given base field, we focus on rational squares that occur in arithmetic progressions, but not necessarily in immediate succession

Summary

Introduction

The following definition dates back to Euler ([1]; see [2]). Definition 1. After factoring out the greatest common divisor of the coefficients, this leads us to consider quadratic forms X 2 − pkY 2 and X 2 + qkY 2 where k, p, q ∈ N with p and q coprime Concordant forms in this form tie up nicely with rational squares occurring in arithmetic progressions. The main difference between our approach and that of other authors such as [3,4,9], (see the section of this paper) is the use of a true isomorphism (which sets up a one-to-one correspondence between nontrivial solutions of the concordant form problem and points of order greater than 2 on the associated curve) rather than a mapping of degree 4 (which causes the loss of solutions associated with 4-torsion points).

Connections to elliptic curves

Nontrivial torsion solutions

Interpretation and conclusions