Solving the Equation (x! x^3 - x): An in-depth Analysis

Introduction

This article explores the equation (x! x^3 - x) and delves into its solutions, including both integer and complex ones. This comprehensive guide uses a mathematical approach to find all possible solutions, backed by computational results using Mathematica.

Step-by-Step Solution Process

Step 1: Rewrite the Equation

The given equation is (x! x^3 - x). We can rewrite it as:

(x! x(x^2 - 1))

This reveals that the right-hand side is a polynomial expression, making it easier to test for integer solutions.

Step 2: Test Integer Values

We test small integer values of (x) to find solutions:

For (x 0): (0! 1) and (0^3 - 0 0) Not a solution. For (x 1): (1! 1) and (1^3 - 1 0) Not a solution. For (x 2): (2! 2) and (2^3 - 2 6) Not a solution. For (x 3): (3! 6) and (3^3 - 3 24) Not a solution. For (x 4): (4! 24) and (4^3 - 4 60) Not a solution. For (x 5): (5! 120) and (5^3 - 5 120) This is a solution: (x 5). For (x 6): (6! 720) and (6^3 - 6 210) Not a solution. For (x 7): (7! 5040) and (7^3 - 7 336) Not a solution.

For (x 7), the factorial function grows much faster than the cubic polynomial, indicating no additional integer solutions beyond (x 5).

Conclusion

The only solution to the equation (x! x^3 - x) is:

(boxed{5})

Additional Insights Using the Gamma Function

In a broader context, the Gamma function ((Gamma(x))) can extend the factorial function to non-integer values of (x), allowing us to explore possible solutions in the complex plane.

The Gamma Function and Complex Solutions

Consider the equation:

(Gamma(x) x^3 - x)

By plotting the curves (text{Re}(Gamma(x)) - x^3x 0) (blue) and (text{Im}(Gamma(x)) - x^3x 0) (brown) in the complex plane, we can identify additional solutions:

For instance, the intersection of a blue and a brown curve at approximately (x -0.376527 0.592061i) represents a complex solution. These complex solutions are distinct from the integer solutions and highlight the rich behavior of the Gamma function in the complex domain.

Final Thoughts

While the integer solution (x 5) is the only admitted solution, exploring the Gamma function provides a broader perspective on the equation (x! x^3 - x). This approach emphasizes the utility of advanced mathematical functions in solving complex equations, ensuring a thorough understanding of the problem.