  Using 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.
