Integer Solutions to the Equation x4 - y4 x3 - y3

Introduction

The equation x4 - y4 x3 - y3 can be explored in terms of its integer solutions. This article aims to detail the process of solving this equation and determining the integer values that satisfy it.

Factorizing the Equation

To solve the equation, we first rearrange it:

x4 - y4 x3 - y3

This can be rewritten as:

x4 - x3 y4 - y3

Which can be further simplified to:

x3(x - 1) y3(y - 1)

Assuming x y

If x y, then the equation simplifies to:

x3(x - 1) x3(x - 1)

Clearly, this is a valid solution. However, let's explore the case where x ≠ y.

Exploring x ≠ y

Assuming x ≠ y, we can rearrange the equation to:

x3 - y3 y4 - x4

Factoring both sides:

(x - y)(x2 xy y2) (y - x)(y2 xy x2)

This simplifies to:

(x - y)(x2 xy y2) -(x - y)(x2 xy y2)

Considering that x - y ≠ 0 (since x ≠ y), we can divide both sides by x - y:

x2 xy y2 -(x2 xy y2)

This results in:

2(x2 xy y2) 0

Therefore:

x2 xy y2 0.

For this equation to hold true, x and y must both be integers. Given that x2 xy y2 > 0 for all integer values other than x y, the only possibilities are:

x 0 and y 0 x 1 and y 0 x 0 and y 1

Conclusion

The integer solutions to the equation x4 - y4 x3 - y3 are:

(0, 0) (1, 0) (0, 1)

Additionally, the equation is always satisfied when x y, providing all integer pairs (x, x).

Related Keywords

integer solutions equation factorization real integers