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.
