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Statement

Corollary 3 (p. 3). Let $a$ and $D\ge3$ be coprime integers. For every $m\ge2$ there are infinitely many $r\in\mathbb N$ such that

$$p_{r+1}\equiv p_{r+2}\equiv\cdots\equiv p_{r+m}\equiv a \bmod D$$

and $p_{r+m}-p_{r+1}\le DC_m$, where $C_m$ is a constant depending only on $m$. Here $p_n$ is the $n$-th smallest prime.

The paper notes (p. 3) that Shiu proved the result without the constraint $p_{r+m}-p_{r+1}\le DC_m$, and that Shiu attributes to Chowla the conjecture that infinitely many pairs of consecutive primes $p_r,p_{r+1}$ lie in the class $a \bmod D$.

Proof pointer

P. 5. Take $k\ge k_m$ and any admissible $\{x+a_j\}_{j=1}^k$ with $a_1<\cdots<a_k$, and put $b_j=Da_j+a$; then $\{x+b_j\}_{j=1}^k$ is admissible and $\gcd(D,b_j)=1$. Theorem 1 with $g=D$ gives $m$ consecutive primes $Dn+h_i$ for infinitely many $n$; they are all $\equiv a \bmod D$ and lie in an interval of length $b_k-b_1=D(a_k-a_1)$.

Read depth

Claims checked: Corollary 3 and the remark on Shiu's theorem were read clause by clause on the page images of the arXiv print, and the proof on p. 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 $g=D$.

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.