The equations of state at room temperature as well as the energies of crystal structures up to pressures exceeding 100 GPa are calculated for Na and K . It is shown that the allowance for generalized gradient corrections (GGA) in the density functional method provides a precision description of the equation of state for Na, which can be used for the calibration of pressure scale. It is established that the close-packed structures and BCC structure are not energetically advantageous at high enough compressions. Sharply non-monotonous pressure dependences of elastic moduli for Na and K are predicted and melting temperatures at high pressures are estimated from various melting criteria. The phase diagram of K is calculated and found to be in good agreement with experiment. 64.30.+t, 64.70.Kb, 71.25.Pi The theoretical and experimental studies of the matter properties at ultra-high pressures arouse a great interest in the connection with the possibility to obtain phases with uncommon properties as well as geophysical and astrophysical applications. As an example, the problem of metallic hydrogen can be mentioned 1 . In the high pressure studies the alkali metals can be conveniently used as model objects. This is due, first, to their high compressibility and, second, to the variety of physical phenomena occurring in their compression and numerous structural and electron phase transitions (see, e.g. 2-9 ). For heavy alkali metals it is the famous s−d isostructural FCC-FCC transition (see, e.g., 10 and references therein) as well as the transitions to uncommon distorted phases at higher pressures 11 . Recently it was supposed, basing on the electron structure calculations, that lithium can transform at high enough pressures into "exotic" phases similar to that of hydrogen 12 . Thus, further theoretical investigations of structural properties of alkali metals at ultra-high pressures seem to be interesting and important.Despite a lot of considerations, this is still an open problem. The most of early attempts used computational approaches which were not accurate enough from the contemporary point of view. It is well known (see, e.g., 13 ) that the highly accurate quantitative description of the electronic and, especially, lattice properties of metals needs the consideration of the real form of potential in the crystal and going beyond the frame of local approximation in the density functional, in particular, the allowance for generalized gradient corrections (GGA) 14 .In the present work a consistent theoretical study of the relative stability of crystal structures of Na and K under pressure as well as a variety of related lattice properties, is performed basing on these first-principle calculations. The most interesting result obtained is that, contrary to the traditional concepts (see, e.g., 9 ) neither structure, which is characteristic of metals under normal conditions (BCC, FCC and HCP), is stable at high enough pressures even in Na where there are no electron transitions.The ab initio calculations of electronic st...