Consecutive primes in tuples

William D. Banks¹ ; Tristan Freiberg¹ ; Caroline L. Turnage-Butterbaugh²

¹ Department of Mathematics University of Missouri Columbia, MO 65211, U.S.A.
² Department of Mathematics University of Mississippi University, MS 38677, U.S.A.

Acta Arithmetica, Tome 167 (2015) no. 3, pp. 261-266

Résumé

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 $a \bmod D$.

Détail

DOI : 10.4064/aa167-3-4

Keywords: stunning advance towards prime k tuple conjecture maynard tao have shown sufficiently large terms admissible k tuple mathcal linear forms mathbb set mathcal contains least primes infinitely many mathbb note deduce mathcal contains least consecutive primes infinitely many mathbb answer old question erd tur producing strings consecutive primes whose successive gaps delta ldots delta form increasing resp decreasing sequence strings exist delta j delta coprime integers arbitrarily long strings consecutive primes bounded gaps congruence class bmod