Integer Solution Patterns in Transcendental Equations with Six Unknowns

Abstract

This study presents an analytical investigation of the transcendental equation and examines its structural properties using appropriate algebraic techniques.

Introduction

A transcendental equation is an equation that contains transcendental functions such as logarithmic, exponential, trigonometric, hyperbolic, or inverse trigonometric functions. These equations cannot usually be solved using simple algebraic methods, so numerical or analytical techniques are often used to obtain approximate or exact solutions. Transcendental equations also have important links with number theory, especially with Diophantine equations, where solutions are restricted to integers. Studying these relationships helps reveal deeper patterns and structures within mathematics.

Transcendental functions are widely used in mathematical analysis and scientific applications. Since these functions cannot be expressed as the roots of polynomial equations with finite operations, they play a unique role in advanced mathematical modeling.

In this study, the properties and solution methods of certain transcendental equations are analyzed. A linear transformation is applied to simplify the given equation and derive an equivalent form. Using this transformed equation, several patterns of integer solutions are investigated.

Four different solution patterns are developed by applying factorization techniques and separating real and imaginary parts of the equations. Each pattern produces a set of non-zero integer solutions expressed in terms of parameters. Numerical examples are provided in tables to verify that the left-hand side equals the right-hand side, confirming the validity of the derived solutions.

Conclusion

In this post, we have explained the essential solutions to the transcendental equation under complex patterns utilizing various numerical examples. Additionally, one could look for detailed answers to these similar types of equations.

References

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Copyright

Copyright © 2026 P. Saranya, B. Narmadha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.