Consecutive primes in tuples

In a recent advance towards the Prime $k$-tuple Conjecture, Maynard and Tao have shown that if $k$ is sufficiently large in terms of $m$, then for an admissible $k$-tuple $\mathcal{H}(x) = \{gx + h_j\}_{j=1}^k$ of linear forms in $\mathbb{Z}[x]$, the set $\mathcal{H}(n) = \{gn + h_j\}_{j=1}^k$ contains at least $m$ primes for infinitely many $n \in \mathbb{N}$. In this note, we deduce that $\mathcal{H}(n) = \{gn + h_j\}_{j=1}^k$ contains at least $m$ consecutive primes for infinitely many $n \in \mathbb{N}$. We answer an old question of Erd\H os and Turán by producing strings of $m + 1$ consecutive primes whose successive gaps $δ_1,\ldots,δ_m$ form an increasing (resp. decreasing) sequence. We also show that such strings exist with $δ_{j-1} \mid δ_j$ for $2 \le j \le m$. For any coprime integers $a$ and $D$ we find arbitrarily long strings of consecutive primes with bounded gaps in the congruence class $a \bmod D$.

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