he Low Energy QCD allows to calculate the $\pi\pi$ and $\pi K$ scattering lengths with high precision. There are accurate relations between these scattering lengths and $\pi^+\pi^-$, $\pi^-K^+$, $\pi^+K^-$ atoms lifetimes. The experiment on the first observation of $\pi^-K^+$ and $\pi^+K^-$ atoms is described. The atoms were generated in Nickel and Platinum targets hit by the PS CERN proton beam with momentum of 24 GeV/$c$. Moving in the target, part of atoms break up producing characteristic $\pi K$ pairs (atomic pairs) with small relative momentum $Q$ in their c.m.s. In the experiment, we detected $n_A=349\pm62$ (5.6 standard deviations) $\pi^-K^+$ and $\pi^+K^-$ atomic pairs. The main part of $\pi K$ pairs are produced in free state. The majority of such particles are generated directly or from short-lived sources as $\rho$, $\omega$ and similar resonances. The electromagnetic interactions in the final state create Coulomb pairs with a known sharp dependence on $Q$. This effect allows to evaluate the number of these Coulomb pairs. There is a precise ratio ($\sim$1\%) between the number of $\pi^-K^+$ ($\pi^+K^-$) Coulomb pairs with small $Q$ and the number of produced $\pi^-K^+$ ($\pi^+K^-$) atoms. Using this ratio, we obtained the numbers of generated $\pi^-K^+$ and $\pi^+K^-$ atoms. The atom breakup probability in a target $P_{\mathrm{br}}=n_A/N_A$ depends on the atom lifetime. Using such dependences for the Ni and Pt targets, known with a precision about 1\%, the $\pi K$ atom lifetime was measured and from this value the $\pi K$ scattering lengths were evaluated. The presented analysis shows that the $\pi^-K^+$ and $\pi^+K^-$ atoms production in the p-nucleus interactions increases by 16 and 38 times respectively if the proton momentum is increased from 24 GeV/$c$ up to 450 GeV/$c$.