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On equivalent forms of the weak Goldbach conjecture

Harald Helfgott has announced a proof of the odd Goldbach conjecture (also known as the ternary or weak Goldbach conjecture). This is big news. Like a good maths newshound, Christian Perfect promptly wrote this up for The Aperiodical as “All odd integers greater than 7 are the sum of three odd primes!

Wait, though, there’s a problem. As Relinde Jurrius pointed out on Twitter, the formulation used in the paper abstract was not quite the same.

The ternary Goldbach conjecture, or three-primes problem, asserts that every odd integer $N$ greater than $5$ is the sum of three primes. The present paper proves this conjecture.

The version Christian used makes the assertion using odd primes, whereas the paper abstract only claims “the sum of three primes”. The latter version includes $7$ because $7$ can be written as the sum of three primes, but not odd ones ($7 = 3+2+2$). Certainly, you can see both statements of the weak Goldbach conjecture used (for example, here’s the $\gt 5$ version and here’s the $\gt 7$ version). Are they equivalent?