We show that strong Luttinger correlations of the electron liquid in armchair carbon nanotubes significantly enhance the onset temperature of the putative twist Peierls instability and lead to its 1/R3 dependence on the tube radius. Depending on the values of the coupling constants the umklapp processes can either assist or compete with the twist instability. In the case of a competition the umklapp processes win in wide tubes. In narrow tubes the outcome of the competition depends on the relative strength of the e-e and e-ph backscattering. Our estimates show that the twist instability may be realized in freestanding (5,5) tubes.