2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/31784Using the cavity equations of \cite{mezard:parisi:zecchina:02,mezard:zecchina:02}, we derive the various threshold values for the number of clauses per variable of the random $K$-satisfiability problem, generalizing the previous results to $K \ge 4$. We also give an analytic solution of the equations, and some closed expressions for these thresholds, in an expansion around large $K$. The stability of the solution is also computed. For any $K$, the satisfiability threshold is found to be in the stable region of the solution, which adds further credit to the conjecture that this computation gives the exact satisfiability threshold.38 pages; extended explanations and derivations; this version is going to appear in Random Structures & AlgorithmsComputational ComplexityDisordered Systems and Neural NetworksDiscrete MathematicsF.2.0; G.2.0text