Integral Solutions for the Diophantine Equation of Higher Degree with Six Unknowns x⁶ − y⁶ − 3456z³ = 800(p² − q²)R⁸

Our focus in this paper has been on solving high-power Diophantine equations - those with variables raised to a high degree. These types of equations can be particularly challenging as they involve finding integer solutions that satisfy the given polynomial equation. We have utilized various techniques such as brute force methods and substitution strategies to solve these high power Diophantine equations, successfully deriving their solutions. Furthermore, our investigation has led us to uncover intriguing relationships among these solutions, which manifest in four distinct patterns. The equation x⁶ − y⁶ − 3456z³ = 800(p² − q²)R⁸ is analysed with properties. Some of the special numbers are discussed in properties. Special numbers are unique and have special qualities that set them apart from other numbers. Learning about these special qualities helps us understand how numbers work and their significance in different areas of math and science.