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Let $A = \{a_1, a_2, \ldots\}$ and $B = \{b_1, b_2, \ldots\}$ be infinite, strictly increasing sequences of natural numbers. Define $S_{ij} = a_i + b_j$.

Question: Do there exist sequences $A$ and $B$ such that $a_i + b_j$ is prime if and only if $i < j$?

In particular, we need:

  • $a_i + b_j$ is prime whenever $i < j$
  • $a_i + b_j$ is composite whenever $j \leq i$
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    $\begingroup$ This is of course relevant. $\endgroup$ Commented Jun 28 at 20:21
  • $\begingroup$ As pointed out by mathworer21, Tao and Ziegler show that there are sequences $(a_i)$ and $(b_j)$ satisfying the first condition $a_i+b_j$ is prime whenever $i<j$. Since $a_i+b_j$ is either a prime or a composite for any $j\leq i$, using Ramsey's theorem one can extract subsequences $(c_i)$ of $(a_i)$ and $(d_j)$ of $(b_j)$ which either satisfy the second condition; or satisfy that $c_i+d_j$ is prime for every $i,j$. $\endgroup$ Commented Jun 29 at 21:38
  • $\begingroup$ Also I expect the answer to be almost certainly yes, and it may even follow from the Tao-Ziegler argument if one is careful enough to keep track of the composite numbers in the Maynard sieve (which should be the easy part). $\endgroup$ Commented Jun 29 at 21:48
  • $\begingroup$ Is there any reason for introducing the notation $S_{ij}$? $\endgroup$ Commented Jun 29 at 22:29
  • $\begingroup$ @YemonChoi: initially was attempting to visualize it an infinite grid where we need to color entries based on their position relative to the diagonal. But then I discarded that viewpoint as it became too complicated. $\endgroup$ Commented Jun 29 at 22:39

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