The Non-Existence of Integer Solutions to The Quartic-Cubic Diophantine Equation Y^3 + XY = X^4 + 4: A Complete Resolution Via Factorization and Modular Arithmetic

Abstract

We establish the complete non-existence of integer solutions to the Diophantine equation Y^3 + XY = X^4 + 4, thereby resolving an open problem in the classification of quartic- cubic Diophantine equations. Our proof employs a novel synthesis of classical techniques: we utilize Sophie Germain’s identity for the factorization of quartic forms, develop a com- prehensive greatest common divisor stratification, and apply systematic modular arithmetic obstructions combined with the unique factorization property in Z.
The proof proceeds through an exhaustive case analysis based on d = gcd(x, y), where we show that d ∈ {1, 2, 4} is necessary, and then demonstrate that each case leads to a polynomial equation with no integer roots. We establish several auxiliary results on the coprimality structure of the factored forms and the impossibility of certain quartic polynomial equations over Z.
Our methods extend beyond this specific equation, providing a template for attacking similar mixed-degree Diophantine problems. We complement our theoretical analysis with rigorous computational verification and propose several generalizations, connecting our re- sult to the broader landscape of Diophantine analysis, including connections to genus-1 curves and the study of integral points on algebraic varieties. The techniques developed herein contribute to the ongoing classification program for Diophantine equations of low degree and small height.