PreprintCC BY 4.0arXiv
A Criterion for classifying Elliptic curve that are unsolvable in the set integers
Abstract
In this work, we establish a criterion for classifying elliptic curves that have no solutions in the set of positive integers. In particular, we prove that if E: y2 = Ax3 + Bx2 + Cx + D is an elliptic curve where A, B, C, and D are coefficients of the curve, and p is a prime number where p ≡ 1 pmod4, and (x,y) is a positive integer point, then if −3A + C ≡ −1 pmod4, A − B + C ≡ −1 pmod4, and p2 = A − B + C − B, then there are no positive solutions for the curve E, where E(ℤ+) = ∅. Furthermore, we generalize this criterion by incorporating techniques from linear algebra. Specifically, we introduce a matrix construction to establish the nonexistence of positive integer solutions for a family of elliptic curves.
Related
- A linear independence criterion for p-adic Hurwitz zeta values at odd integers
- Algebra of the Infrared with Elliptic Curve-Valued Potential
- Counterexamples to Dujella's conjecture on integral points on the elliptic curve attached to a Diophantine triple
- Elliptic curve rank growth in multi-quadratic extensions
- Methods to Find Integer Points on the Elliptic Curve of Factorizable Polynomial Equation
BibTeX
@misc{fadul2026criterion,
title = {{A Criterion for classifying Elliptic curve that are unsolvable in the set integers}},
author = {Shazali Abdalla Fadul},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22344},
eprint = {2609.22344},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22344},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Criterion for classifying Elliptic curve that are unsolvable in the set integers
AU - Shazali Abdalla Fadul
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.22344
UR - https://arxiv.org/abs/2609.22344
AB - In this work, we establish a criterion for classifying elliptic curves that have no solutions in the set of positive integers. In particular, we prove that if E: y^2 = Ax^3 + Bx^2 + Cx + D is an elliptic curve where A, B, C, and D are coefficients of the curve, and p is a prime number where p ≡ 1 pmod4, and (x,y) is a positive integer point, then if −3A + C ≡ −1 pmod4, A − B + C ≡ −1 pmod4, and p^2 = A − B + C − B, then there are no positive solutions for the curve E, where E(ℤ^+) = ∅. Furthermore, we generalize this criterion by incorporating techniques from linear algebra. Specifically, we introduce a matrix construction to establish the nonexistence of positive integer solutions for a family of elliptic curves.
ER -