A Lorentzian hypersurface M^4_1 of Minkowski 5−space (i.e. E^5_1), defined by an isometric immersion x : M^4_1 → E_1^5, is said to be L_k-biconservative if the tangent component of (L_k)^2 x is identically zero, where L_k is the kth extension of Laplace operator ∆ = L_0. The operator Lk is the linearized operator arisen from the first variation of (k + 1)th mean curvature vector field on M^4_1 . This subject is motivated by a well-known conjecture of Bang-Yen Chen which says that the condition ∆^2 x = 0 implies the minimality for submanifolds of Euclidean spaces. In this paper, we study Lk-biconservative Lorentzian hypersurfaces of E_1^5 in four different cases based on the matrix representation forms of the shape operator. We show that if such a hypersurface has constant mean curvature and at most two distinct principal curvatures, then its (k + 1)th mean curvature is constant.
