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Statement

Setting (p. 3). A Polignac number is a positive even number $2k$ with $p_{n+1}-p_n=2k$ for infinitely many $n$ (Definition 1, p. 3; see Theorem 1).

Theorem 2 (p. 3, quoted). "There exists an ineffective constant $C'$ such that every interval of type $[M,M+C']$ contains at least one Polignac number."

Polignac numbers here are strong Polignac numbers (Remark, p. 3).

Proof pointer

Pages 9--10. By contradiction: if the theorem fails, there are intervals $I_\nu=[M_\nu,M_\nu+C_\nu]$ with $M_\nu>C_\nu>4M_{\nu-1}$ and $M_1>C_0$ that contain no Polignac number (4.4)--(4.5). Choose an admissible $k$-tuple with $h_\nu\in[M_\nu+C_\nu/2,M_\nu+C_\nu]$, $k\ge k_0$; every difference $h_\mu-h_\nu$, $\nu<\mu$, then lies in $I_\mu$ (4.8). The Main Theorem makes some such difference a gap between consecutive primes infinitely often (the paper writes "can be written as a difference of two consecutive primes", p. 10), a contradiction. The paper notes that the resulting constant is ineffective (p. 9).

Read depth

Claims checked: the statement was read clause by clause on the printed pages of arXiv:1305.6289v1, and the proof on pp. 9--10 was followed. Nothing here is independently reviewed.

Dependencies

The Main Theorem (p. 6) of this paper.

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

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