Polar optical phonon modes in wurtzite GaN nanowires with equilateral triangular cross-section are deduced and analyzed by using the dielectric continuum model. It is found that the frequency of exactly confined (EC) modes in the GaN nanowire systems is the characteristic frequency (ωz,L) of longitudinal-optical phonons in free z-direction, which is obviously different from the case of EC modes in wurtzite GaN-based quantum wells due to their different confined dimensions. Then by using the method of nonseparable variables, the Laplace equation of the electrostatic potential of the nanowire structure is solved, the exact analytical phonon states of EC modes are obtained, and the polarization eigenvectors and their orthogonal relations, the free phonon field, as well as the corresponding Frohlich electron-phonon interaction Hamiltonian for the EC phonon modes are also derived. Finally, the numerical calculation on GaN nanowire is performed, the electron-phonon coupling functions are depicted and discussed, the symmetries and coupling strength of the coupling functions are analyzed in detail, and some meaningful conclusions are drawn.