On $\theta$-congruent numbers, rational squares in arithmetic progressions, concordant forms and elliptic curves

TL;DR

This paper extends classical correspondences involving rational right triangles, squares in arithmetic progression, and quadratic forms to generalized concepts using elliptic curves, clarifying existing results and addressing open questions.

Contribution

It introduces a unified framework connecting rational $ heta$-triangles, squares in arithmetic progressions, and concordant forms via elliptic curves, expanding previous theories.

Findings

Established one-to-one mappings to rational points on elliptic curves.

Analyzed the role of solutions to the $ heta$-congruent number problem.

Extended and unified results on quadratic forms and elliptic curves.

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 $\theta$-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 $\theta$-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.