The eigenvalues λ_n and eigenvectors f_n of the 5D discrete number operator N_5=A_5^⊺ A_5 are evaluated in a systematic way. Because the eigenvalues λ_n are represented by distinct non-negative numbers, the number operator N_5 has been used to classify eigenvectors of the 5D discrete Fourier transform Φ_5, thus resolving the ambiguity caused by the well-known degeneracy of the eigenvalues of the discrete Fourier transform Φ_N. A procedure for sparsealization the intertwining operators A_5 and A_5^⊺ has been formulated, which made it possible to construct a discrete analog of the continuous-case formula ψ_n (x)=1/√n! (a^† )^n ψ_0 (x). In addition, a discrete analog for the eigenvectors f_n of the another continuous-case formula ψ_n (x)=c_n^(-1) H_n (x) ψ_0 (x),  c_n=√(2^n n!), has been established in terms of the Newtonian basis polynomials P_n (X_5 ), n∈Z_5, times the lowest eigenvector f_0.