The parity problem obstruction for the binary Goldbach problem with bounded error

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Two of the most famous open problems in additive prime number theory are the twin prime conjecture and the binary Goldbach conjecture. They have quite similar forms:

  • Twin prime conjecture The equation $latex {p_1 – p_2 = 2}&fg=000000$ has infinitely many solutions with $latex {p_1,p_2}&fg=000000$ prime.
  • Binary Goldbach conjecture The equation $latex {p_1 + p_2 = N}&fg=000000$ has at least one solution with $latex {p_1,p_2}&fg=000000$ prime for any given even $latex {N geq 4}&fg=000000$.

In view of this similarity, it is not surprising that the partial progress on these two conjectures have tracked each other fairly closely; the twin prime conjecture is generally considered slightly easier than the binary Goldbach conjecture, but broadly speaking any progress made on one of the conjectures has also led to a comparable amount of progress on the other. (For instance, Chen’s theorem has a version for the twin prime conjecture, and a version…

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