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Mathematics 2015, 3(1), 2-15; doi:10.3390/math3010002

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

1
Fachbereich 2, Fachhochschule Frankfurt, Nibelungenplatz 1, D-60318 Frankfurt am Main, Germany
2
Fachbereich Architektur und Bauingenieurwesen, Studiengang Angewandte Mathematik, Hochschule RheinMain, Kurt-Schumacher-Ring 18, D-65197 Wiesbaden, Germany
*
Author to whom correspondence should be addressed.
Academic Editor: Ken Ono
Received: 28 October 2014 / Revised: 11 December 2014 / Accepted: 8 January 2015 / Published: 19 January 2015
View Full-Text   |   Download PDF [232 KB, uploaded 19 January 2015]

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. View Full-Text
Keywords: elliptic curves; concordant forms; θ-congruent numbers elliptic curves; concordant forms; θ-congruent numbers
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Selder, E.; Spindler, K. On θ-Congruent Numbers, Rational Squares in Arithmetic Progressions, Concordant Forms and Elliptic Curves. Mathematics 2015, 3, 2-15.

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