prime k-tuple conjecture

《Algebraic Number Theory》 · 第 224 页 · Section 7.8，Prime k-tuple conjecture

Let $a_1,a_2,\ldots,a_k$ be given integers. Then there exist infinitely many positive integers $n$ such that $n+a_1,n+a_2,\ldots,n+a_k$ are all prime, provided that for every prime number $p$ there is an integer $n$ such that $(n+a_i,p)=1$ for $i=1,2,\ldots,k$.