we think it mainly technical to get it for all $k \ge 3.$)

\[mod3\] Let $F(n, p)$ be the random set of equations $\mod 3:$ $ x_1 + \dots + x_k \,=\, a , \, 0 \le a \le 2,$ $x_1+ \dots  +x_k$ an ordered $k-tuple$ of variables. If $p < (T-\varepsilon)/n^{k-1}$ $F(n, p)$ is satisfiable whp. for $k>15.$

The main task is to show that a $2-$core of density $\le 1-\varepsilon$ has a solution with probability $>\varepsilon>0$ . Our proof starts as Dubois/Mandler: Let $X$ be the number of satisfying assignments of the $2-$core. Its expectation is $ \ge d^{\varepsilon n}, d=3.$ We show that $E[X^2] \le O(E[X])^2.$ This implies (by Cauchy-Schwartz (or Paley-Zygmund) inequality) that the probability to have a solution is $\ge \varepsilon >0.$ By Fact \[Core\] $F(n, p)$ has a solution with the same probability. We apply Friedgut-Bourgain’s Theorem to $F(n, p)$ to show that unsatisfiability has a sharp threshold. By this the probability becomes $1-o(1).$ In [@CRDA] Friedgut-Bourgain is applied to the $\mod 2-$case. It seems that our proof for the $\mod 3-$case is somewhat simpler (and applies to the $\mod 2-$case and other cases.)

To determine $E[X^2]$ Dubois/Mandler apply Laplace Method (one ingredient: bounding 