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

Abstract

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 this paper, 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.