The idea of prime pairs is extended to prime $k$-tuplets (e.g. $p$, $p+2$, $p+6$ is a pattern for a 3-tuplet) and all possible patterns are obtained for $k\leq 20$. It is not known if patterns exist for every $k$. The largest known primes for each pattern are given for $k\leq 16$ (and for one pattern, with $k=17$, in an addendum). The method of search for $k=16$ is described in detail. The method for finding prime 16-tuplets can be used for investigating Cunningham chains (sequences of primes such that $p_{i+1}= 2p_i+1$ or $p_{i+1}= 2p_i-1$) and a number of results are reported.
Reviewer: H.J.Godwin (Egham)