Fruit Diophantine Equation

TL;DR

This paper proves that a specific Diophantine equation has no integral solutions and as a result, certain elliptic curves also lack integral points, advancing understanding in number theory.

Contribution

It establishes the non-existence of integral solutions for a particular Diophantine equation and related elliptic curves, providing new insights into their properties.

Findings

The Diophantine equation X^3 + XYZ = Y^2 + Z^2 + 5 has no integral solutions.

The family of elliptic curves Y^2 - kXY = X^3 - (k^2 + 5) has no integral points.

The results contribute to the theory of Diophantine equations and elliptic curves.

Abstract

We show that the Diophantine equation given by X^3+ XYZ = Y^2+Z^2+5 has no integral solution. As a consequence, we show that the family of elliptic curve given by the Weierstrass equations Y^2-kXY = X^3 - (k^2+5) has no integral point.