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Statement

Setting (p. 1). A $k$-tuple of linear forms $\mathcal H(x)=\{g_jx+h_j\}_{j=1}^k$ in $\mathbb Z[x]$ is admissible when the polynomial $\prod_{j=1}^k(g_jx+h_j)$ has no fixed prime divisor, that is, for every prime $p$ the number of residues $n \bmod p$ at which it vanishes mod $p$ is less than $p$. The paper considers only tuples with $g_1,\ldots,g_k>0$ and $\prod_{1\le i<j\le k}(g_ih_j-g_jh_i)\ne0$, its condition (1).

The input (p. 1) is the Maynard-Tao theorem in Granville's formulation, which the paper quotes and does not prove: for every $m\in\mathbb N$ with $m\ge2$ there is $k_m$, depending only on $m$, such that for every integer $k\ge k_m$ and every admissible $\{g_jx+h_j\}_{j=1}^k$ satisfying (1), the set $\{g_jn+h_j\}_{j=1}^k$ contains $m$ primes for infinitely many $n\in\mathbb N$; one may take any $k_m$ with $k_m\log k_m>e^{8m+4}$.

Theorem 1 (p. 2). Let $m,k\in\mathbb N$ with $m\ge2$ and $k\ge k_m$, $k_m$ as in the Maynard-Tao theorem. Let $b_1,\ldots,b_k$ be distinct integers with $\{x+b_j\}_{j=1}^k$ admissible, and let $g$ be a positive integer coprime to $b_1\cdots b_k$. Then there is a subset $\{h_1,\ldots,h_m\}\subseteq\{b_1,\ldots,b_k\}$ such that, for infinitely many $n\in\mathbb N$, the numbers $gn+h_1,\ldots,gn+h_m$ are consecutive primes.

The paper notes (p. 2) that the case $m=2$, $g=1$, with the weaker bound $k_2\ge3.5\times10^6$, was proved earlier by Pintz by a different argument.

Proof pointer

Pp. 3-4. After shifting so that $1<b_1<\cdots<b_k$, each integer $t$ in $[1,b_k]$ that is not a $b_j$ is assigned its own prime $q_t$, coprime to $g$ and with $t\not\equiv b_j \bmod q_t$ for every $j$; the Chinese remainder theorem gives $a$ with $ga+t\equiv0 \bmod q_t$ for all such $t$. With $Q=\prod q_t$, the tuple $\{gQx+ga+b_j\}$ is admissible and satisfies (1), and every prime in $[g(QN+a)+b_1,\,g(QN+a)+b_k]$ is one of its values. Taking the largest $m'$ such that some $m'$ of these forms are simultaneously prime for infinitely many $N$, the Maynard-Tao theorem gives $m'\ge m$, and maximality forces the remaining forms to be composite for all large such $N$, so those $m'$ primes are consecutive.

Read depth

Claims checked: the definitions, the quoted Maynard-Tao statement and Theorem 1 were read clause by clause on the page images of the arXiv print, and the proof on pp. 3-4 was followed. The Maynard-Tao theorem is cited, not proved, in the paper and was not checked here. Nothing here is independently reviewed.

Dependencies

None in the corpus. External input: the Maynard-Tao theorem (Maynard, Small gaps between primes, Ann. of Math. (2) 181 (2015); Granville's formulation, Theorem 6.2 of Primes in intervals of bounded length); Maynard's paper has its own library card.

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 theorem is the step from which the paper's Corollary 1 is deduced; on its own it orders no gaps.