We explore congruent numbers through a unified approach combining geometric constructions, elliptic curve theory, and arithmetic progressions of squares. Leveraging recent advances in modular forms and computational techniques, we construct infinite explicit families of congruent numbers parameterized by Pell-type equations and related Diophantine conditions. We complement this with a detailed statistical analysis of the residue classes, rank distributions, and root numbers associated with these families, providing empirical insights that deepen understanding of intricate conjectures in the arithmetic of elliptic curves.
| Published in | American Journal of Applied Mathematics (Volume 13, Issue 5) |
| DOI | 10.11648/j.ajam.20251305.16 |
| Page(s) | 360-364 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2025. Published by Science Publishing Group |
Congruent Numbers, Elliptic Curves, Arithmetic Progressions, Pythagorean Triples, Rank, Parametric Families
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APA Style
Vincent, K. K. (2025). Unified Approaches to Congruent Numbers via Geometry, Elliptic Curves and Arithmetic Progression of Squares. American Journal of Applied Mathematics, 13(5), 360-364. https://doi.org/10.11648/j.ajam.20251305.16
ACS Style
Vincent, K. K. Unified Approaches to Congruent Numbers via Geometry, Elliptic Curves and Arithmetic Progression of Squares. Am. J. Appl. Math. 2025, 13(5), 360-364. doi: 10.11648/j.ajam.20251305.16
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Download@article{10.11648/j.ajam.20251305.16,
author = {Kouakou Kouassi Vincent},
title = {Unified Approaches to Congruent Numbers via Geometry, Elliptic Curves and Arithmetic Progression of Squares
},
journal = {American Journal of Applied Mathematics},
volume = {13},
number = {5},
pages = {360-364},
doi = {10.11648/j.ajam.20251305.16},
url = {https://doi.org/10.11648/j.ajam.20251305.16},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251305.16},
abstract = {We explore congruent numbers through a unified approach combining geometric constructions, elliptic curve theory, and arithmetic progressions of squares. Leveraging recent advances in modular forms and computational techniques, we construct infinite explicit families of congruent numbers parameterized by Pell-type equations and related Diophantine conditions. We complement this with a detailed statistical analysis of the residue classes, rank distributions, and root numbers associated with these families, providing empirical insights that deepen understanding of intricate conjectures in the arithmetic of elliptic curves.
},
year = {2025}
}
TY - JOUR T1 - Unified Approaches to Congruent Numbers via Geometry, Elliptic Curves and Arithmetic Progression of Squares AU - Kouakou Kouassi Vincent Y1 - 2025/10/22 PY - 2025 N1 - https://doi.org/10.11648/j.ajam.20251305.16 DO - 10.11648/j.ajam.20251305.16 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 360 EP - 364 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20251305.16 AB - We explore congruent numbers through a unified approach combining geometric constructions, elliptic curve theory, and arithmetic progressions of squares. Leveraging recent advances in modular forms and computational techniques, we construct infinite explicit families of congruent numbers parameterized by Pell-type equations and related Diophantine conditions. We complement this with a detailed statistical analysis of the residue classes, rank distributions, and root numbers associated with these families, providing empirical insights that deepen understanding of intricate conjectures in the arithmetic of elliptic curves. VL - 13 IS - 5 ER -
Applied Fundamental Sciences Department, Nangui Abrogoua University, Abidjan, Côte d’Ivoire
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