Consecutive primes in tuples
Tom 167 / 2015
Streszczenie
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 $a \bmod D$.
