Year: 2026
Type: preprint
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 $E: y^2 = Ax^3 + Bx^2 + Cx + D$ is an elliptic curve where $A$, $B$, $C$, and $D$ are coefficients of the curve, and $p$ is a prime number where $p \equiv 1 \pmod{4}$, and $(x,y)$ is a positive integer point, then if $$ -3A + C \equiv -1 \pmod{4}, $$ $$ A - B + C \equiv -1 \pmod{4}, $$ and $$ p^2 = A - B + C - B, $$ then there are no positive solutions for the curve $E$, where $E(\mathbb{Z}^{+}) = \varnothing$. 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.
Source: arXiv (Cornell University)
Author: Fadul, Shazali Abdalla
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214021526