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

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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