  Let $k\ge 1$ be an integer, and let $P= (f_1(x), \ldots, f_k(x) )$ be $k$
admissible linear polynomials over the integers, or \textit{the pattern}. We
present two algorithms that find all integers $x$ where $\max{ \{f_i(x) \} }
\le n$ and all the $f_i(x)$ are prime.
  Our first algorithm takes at most $O_P(n/(\log\log n)^k)$ arithmetic
operations using $O(k\sqrt{n})$ space.
  Our second algorithm takes slightly more time, $O_P(n/(\log \log n)^{k-1})$
arithmetic operations, but uses only $n^{1/c}$ space for a constant $c>2$. We
prove correctness unconditionally, but the running time relies on two unproven
but reasonable conjectures.
  We are unaware of any previous complexity results for this problem beyond the
use of a prime sieve. We also implemented several parallel versions of our
second algorithm to show it is viable in practice. In particular, we found some
new Cunningham chains of length 15, and we found all quadruplet primes up to
$10^{17}$.
