## A general parity problem obstruction

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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