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arXiv · 2609.22344

A Criterion for classifying Elliptic curve that are unsolvable in the set integers

Shazali Abdalla Fadul
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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.

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Shazali Abdalla Fadul. 2026-09-16. A Criterion for classifying Elliptic curve that are unsolvable in the set integers. https://arxiv.org/abs/2609.22344

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