2014/08/07 by Erich Selder, Selder, Erich, Karlheinz Spindler +1
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1408.1522
openalex publication_date 2014/08/07 · arxiv created 2014/08/22 · arxiv updated 2014/08/25 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
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.