## Consecutive primes in tuples

### Volume 167 / 2015

#### Abstract

In a stunning new 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 Erdős and Turán by producing
strings of $m + 1$ consecutive primes whose successive gaps
$
\delta_1,\ldots,\delta_m
$
form an increasing (resp. decreasing) sequence.
We also show that such strings exist with
$
\delta_{j-1} {\,|\,} \delta_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 $a \bmod D$.