On Some Mixed Polynomial Exponential Diophantine Equation: \(α^n+β^n+a(α^s±β^s)^m+D=r(u^k+v^k+w^k)\) with \(α\) and \(β\) Consecutive

Let \(a,α,β,r,u,v,w\) and \(D\) be any integers and suppose that \(n,m,s\) and \(k\) are non-negative exponent. In thispaper, the diophantine equation \(α^n+β^n+a(α^s±β^s)^m+D=r(u^k+v^k+w^k)\) is developed and investigated for integer solution and its various polynomial identities. Moreover, the study formulates some conjectures for the title equation.