Two algorithms to find primes in patterns

Abstract

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 $10^{17}$.