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Statement

Setting (p. 2). Write $p_n$ for the $n$-th smallest prime and $d_n=p_{n+1}-p_n$. A sequence $(\delta_j)_{j=1}^m$ of positive integers is a run of consecutive prime gaps when, for some natural number $r$, $\delta_j=d_{r+j}=p_{r+j+1}-p_{r+j}$ for $1\le j\le m$.

Corollary 1 (p. 2). For every $m\ge2$ there are infinitely many runs $(\delta_j)_{j=1}^m$ of consecutive prime gaps with $\delta_1<\cdots<\delta_m$, and infinitely many with $\delta_1>\cdots>\delta_m$.

The stronger runs (p. 2, the paragraph after Corollary 1; built in the proof on p. 4). The proof gives infinitely many runs with

$$\delta_1+\cdots+\delta_{j-1}<\delta_j\qquad(2\le j\le m),$$

and infinitely many with

$$\delta_j>\delta_{j+1}+\cdots+\delta_m\qquad(1\le j\le m-1).$$

The paper presents Corollary 1 as answering an old question of Erdős and Turán (their 1948 paper in Bull. Amer. Math. Soc. 54), with pointers to Erdős's paper of the same year and to Guy's Unsolved problems, A11 (p. 2).

Proof pointer

P. 4. Take $k\ge k_{m+1}$ and the admissible tuple $\{x+2^j\}_{j=1}^k$. Theorem 1 (with $g=1$ and $m+1$ in place of $m$) gives exponents $1\le\nu_1<\cdots<\nu_{m+1}\le k$ such that the $n+2^{\nu_j}$ are $m+1$ consecutive primes for infinitely many $n$, so the gaps are $\delta_j=2^{\nu_{j+1}}-2^{\nu_j}$. Then $\delta_1+\cdots+\delta_{j-1}=2^{\nu_j}-2^{\nu_1}<\delta_j$. The decreasing runs come from the tuple $\{x-2^j\}_{j=1}^k$.

Read depth

Claims checked: the definition of a run, Corollary 1 and the stronger runs were read clause by clause on the page images of the arXiv print, and the proof on p. 4 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

• Problem 6: the case $m=3$ of the increasing runs gives infinitely many $r$ with $d_{r+1}<d_{r+2}<d_{r+3}$, so the problem's question has the answer yes.
• Problem 455: the increasing runs give, for every $m$, infinitely many strings of $m+1$ consecutive primes whose gaps strictly increase. The problem asks about the growth of an infinite sequence of primes with non-decreasing gaps, on which the paper says nothing.