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This question came to my mind this afternoon while trying to figure out a possible way to tackle de Polignac's conjecture, which states that every even positive integer can be written as the difference of two consecutive primes in infinitely many ways, and is of course related to About a possible generalization of Green-Tao's theorem.

So here it comes: Can the proof of Green-Tao's theorem be quite naturally slightly modified to show unconditionally that the sequence of $1$-central numbers, i.e positive integers equal to half the sum of two consecutive odd primes, contains arbitrarily long arithmetic progressions?

Many thanks in advance.

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  • $\begingroup$ Seems very unlikely. The G-T theorem essentially relies on lower bounds for prime numbers in an interval. To know that a number is 1-central, you need upper bounds as well - you need to know that $n\pm i$ is not prime for $0<i<k$ (an upper bound on the characteristic function of primes) and the $n\pm k$ is prime (a lower bound). $\endgroup$ – Anthony Quas Sep 29 '15 at 16:24
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    $\begingroup$ Questions like "can proof X be modified to prove statement Y" are not too fortunate in general. Just ask: "Is statement Y true?" Then, as a side remark, you can mention that perhaps proof X can be modified to prove statement Y. $\endgroup$ – GH from MO Sep 29 '15 at 17:07
  • $\begingroup$ @AnthonyQuas: Actually, the statement is true, see my response. $\endgroup$ – GH from MO Sep 29 '15 at 17:33
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The statement in the original post is true. This follows from a recent result of Pintz, which is a common generalization of the Green-Tao theorem and the Maynard-Tao theorem:

Theorem. Let $m\geq 2$ be an integer. Then for any sufficiently large admissible set $\mathcal{H}$, there exist $h_1,\dots,h_m\in\mathcal{H}$ such that the set $$\{n\in\mathbb{N}:\ \text{$n+h_1,\dots,n+h_m$ are consecutive primes}\}$$ contains arbitrary long arithmetic progressions.

Indeed, the special case $m=2$ is already sufficient to derive the statement in the original post.

P.S. Note that the main theorems in Pintz's paper are stated with a typo: the requirement that $p_i^*$ is prime should be dropped (it is the $n$ above). I notified János Pintz, and he agreed with this correction.

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