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Statement

Runs of consecutive prime gaps $(\delta_j)_{j=1}^m$ are as defined on p. 2: $\delta_j=p_{r+j+1}-p_{r+j}$ for $1\le j\le m$ and some natural number $r$, where $p_n$ is the $n$-th smallest prime (see Corollary 1).

Corollary 2 (p. 3). For every $m\ge2$ there are infinitely many runs $(\delta_j)_{j=1}^m$ of consecutive prime gaps with $\delta_{j-1}\mid\delta_j$ for $2\le j\le m$, and infinitely many with $\delta_{j+1}\mid\delta_j$ for $1\le j\le m-1$.

The stronger runs (p. 3, the paragraph after Corollary 2; built in the proof on pp. 4-5). The proof gives infinitely many runs with $\delta_1\cdots\delta_{j-1}\mid\delta_j$ for $2\le j\le m$, and infinitely many with $\delta_m\delta_{m-1}\cdots\delta_{j+1}\mid\delta_j$ for $1\le j\le m-1$.

Proof pointer

Pp. 4-5. Take $k\ge k_{m+1}$ and $Q=\prod_{p\le k}p$, and set $b_1=0$, $b_2=Q$, $b_3=2Q$ and, for $j\ge3$, $b_j=b_{j-1}+\prod_{1\le s<t\le j-1}(b_t-b_s)$. Consecutive differences of this sequence each divide every later one (the paper's (5)), and $\{x+b_j\}_{j=1}^k$ is admissible because $Q$ divides every $b_j$. Theorem 1 with $m+1$ primes gives gaps $\delta_j=b_{\nu_{j+1}}-b_{\nu_j}$; the product of the earlier gaps divides the product of all differences $b_t-b_s$ with $s<t\le\nu_j$, which is $b_{\nu_j+1}-b_{\nu_j}$, and by (5) that divides $\delta_j$. The reversed runs come from $\{x-b_j\}_{j=1}^k$.

Read depth

Claims checked: Corollary 2 and the stronger runs were read clause by clause on the page images of the arXiv print, and the proof on pp. 4-5 was followed. It rests on Theorem 1 and through it on the Maynard-Tao theorem, which the paper cites. Nothing here is independently reviewed.

Dependencies

• Theorem 1 of this paper, applied with $m+1$ primes.

Source. W. D. Banks, T. Freiberg and C. L. Turnage-Butterbaugh, Consecutive primes in tuples, Acta Arith. 167 (2015), no. 3, 261-266, doi:10.4064/aa167-3-4, arXiv:1311.7003; the edition read and its page numbering are named on the source card.

Bears on

No Erdős problem in the corpus is recorded as concerning this corollary.