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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?

Now, clearly if you can prove “every odd integer greater than $7$ is the sum of three odd primes” and point out that “$7=3+2+2$”, then you have “every odd integer greater than $5$ is the sum of three primes”. That, however, isn’t what the abstract claims. The question is, does the implication go the other way?

There are some cases where an odd number can be expressed as three primes including two $2$s (e.g. $15=11+2+2$ or $104733=104729+2+2$), but we can write these as three odd primes without the use of a $2$ (e.g. $15=3+5+7$ or $104733=104723+3+7$). Can we always do this?

We had a useful conversation about this on Google+, and Evelyn Lamb mentioned this on her AMS blog. I also emailed Harald Helfgott and he was kind enough to take the time to reply and post a comment on the Google+ feed.

“People”, Harald implores, “please don’t worry”. It turns out we’d missed him addressing this in the original paper. He acknowledges, on page 114, that “some prefer to state the ternary Goldbach conjecture as follows: every odd number $\ge 9$ is the sum of three odd primes”, and takes account of this when making his conclusion. In doing so, he says in the comment on Google+, “I prove both versions of the conjecture, at the cost of about three lines of extra work”.

In his email he explains that indeed only one version implies the other, so they are not quite equivalent, but that this difference doesn’t matter in practice since this proof, and all previous work on the matter, works equally well for both versions of the conjecture.

So be happy people, both slightly different versions are proven by Helfgott’s work, and we needn’t publish the first ever Aperiodical correction!

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