A general parity problem obstruction

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Many problems and results in analytic prime number theory can be formulated in the following general form: given a collection of (affine-)linear forms $latex {L_1(n),dots,L_k(n)}&fg=000000$, none of which is a multiple of any other, find a number $latex {n}&fg=000000$ such that a certain property $latex {P( L_1(n),dots,L_k(n) )}&fg=000000$ of the linear forms $latex {L_1(n),dots,L_k(n)}&fg=000000$ are true. For instance:

  • For the twin prime conjecture, one can use the linear forms $latex {L_1(n) := n}&fg=000000$, $latex {L_2(n) := n+2}&fg=000000$, and the property $latex {P( L_1(n), L_2(n) )}&fg=000000$ in question is the assertion that $latex {L_1(n)}&fg=000000$ and $latex {L_2(n)}&fg=000000$ are both prime.
  • For the even Goldbach conjecture, the claim is similar but one uses the linear forms $latex {L_1(n) := n}&fg=000000$, $latex {L_2(n) := N-n}&fg=000000$ for some even integer $latex {N}&fg=000000$.
  • For Chen’s theorem, we use the same linear forms $latex {L_1(n),L_2(n)}&fg=000000$ as in the previous two cases, but now…

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