Investigating Computational Generalization of Mixed Polynomial Exponential on Diophantine Equations: αn + βn + α(αsψβs)m + D = r(uk + vk + wk) with consecutive α and β

Let a, α, β, r, u, v, w, and D be integers, and let n, m, s, and k be non-negative exponents. This paper investigates the Diophantine equation αⁿ + βⁿ + a(αˢ + βˢ)ᵐ + D = r(uᵏ + vᵏ + wᵏ) for integer solutions and explores associated polynomial identities. Additionally, several conjectures are formulated in relation to this equation, aiming to extend the understanding of its integer solutions and structural properties. The focus is on finding integer solutions and examining the underlying polynomial identities. The structure of this equation—combining additive powers, multiplicative transformations, and polynomial expressions in multiple integer variables—presents unique challenges and opportunities in the search for integer solutions. By exploring cases for specific values of n, m, s, and k, we develop insights into solution sets, identify relationships among terms, and analyze the symmetry properties inherent in this equation.