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Statement

Theorem 6 (p. 5, quoted). "There is a $d\le7\times10^7$ such that there are arbitrarily long arithmetic progressions of primes with the property that $p'=p+d$ is the prime following $p$ for each element of the progression."

The paper presents it as a common generalization of the Green--Tao theorem and Zhang's bounded gaps (pp. 5, 12), unconditional where the author's 2010 version assumed a level of distribution $\vartheta>1/2$.

Proof pointer

Page 12, by reference only: the Main Theorem is fed into the machinery of Section 7 of the author's 2010 paper (Pintz, Are there arbitrarily long arithmetic progressions in the sequence of twin primes?, Bolyai Soc. Math. Stud. 21), with Green and Tao's method. The argument is not written out in this paper.

Read depth

Claims checked: the statement was read clause by clause on the printed pages of arXiv:1305.6289v1. The proof is by reference to a work that was not read. Nothing here is independently reviewed.

Dependencies

The Main Theorem (p. 6) of this paper; external: Section 7 of the author's 2010 paper, and Green and Tao, The primes contain arbitrarily long arithmetic progressions.

Source. János Pintz, Polignac numbers, conjectures of Erdős on gaps between primes, arithmetic progressions in primes, and the bounded gap conjecture, arXiv:1305.6289v1 (2013); published in From Arithmetic to Zeta-Functions, Springer (2016), 367--384, doi:10.1007/978-3-319-28203-9_22. Labels and pages here are those of arXiv v1. The edition read is named on the source card.

Bears on

None recorded.