Consecutive primes in tuples

William D. Banks, Tristan Freiberg, and Caroline L. Turnage-Butterbaugh. "Consecutive primes in tuples." Acta Arithmetica 167.3 (2015): 261-266. <http://eudml.org/doc/279690>.

@article{WilliamD2015,
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 $(x) = \{gx + h_j\}_\{j=1\}^k$ of linear forms in ℤ[x], the set $(n) = \{gn + h_j\}_\{j=1\}^k$ contains at least m primes for infinitely many n ∈ ℕ. In this note, we deduce that $(n) = \{gn + h_j\}_\{j=1\}^k$ contains at least m consecutive primes for infinitely many n ∈ ℕ. We answer an old question of Erdős and Turán by producing strings of m + 1 consecutive primes whose successive gaps $δ_1,...,δ_m$ form an increasing (resp. decreasing) sequence. We also show that such strings exist with $δ_\{j-1\} | δ_j$ for 2 ≤ j ≤ m. For any coprime integers a and D we find arbitrarily long strings of consecutive primes with bounded gaps in the congruence class a mod D.},
author = {William D. Banks, Tristan Freiberg, Caroline L. Turnage-Butterbaugh},
journal = {Acta Arithmetica},
keywords = {consecutive primes in tuples; bounded gaps between primes; Maynard-Tao theorem},
language = {eng},
number = {3},
pages = {261-266},
title = {Consecutive primes in tuples},
url = {http://eudml.org/doc/279690},
volume = {167},
year = {2015},
}

TY - JOUR
AU - William D. Banks
AU - Tristan Freiberg
AU - Caroline L. Turnage-Butterbaugh
TI - Consecutive primes in tuples
JO - Acta Arithmetica
PY - 2015
VL - 167
IS - 3
SP - 261
EP - 266
AB - 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 $(x) = {gx + h_j}_{j=1}^k$ of linear forms in ℤ[x], the set $(n) = {gn + h_j}_{j=1}^k$ contains at least m primes for infinitely many n ∈ ℕ. In this note, we deduce that $(n) = {gn + h_j}_{j=1}^k$ contains at least m consecutive primes for infinitely many n ∈ ℕ. We answer an old question of Erdős and Turán by producing strings of m + 1 consecutive primes whose successive gaps $δ_1,...,δ_m$ form an increasing (resp. decreasing) sequence. We also show that such strings exist with $δ_{j-1} | δ_j$ for 2 ≤ j ≤ m. For any coprime integers a and D we find arbitrarily long strings of consecutive primes with bounded gaps in the congruence class a mod D.
LA - eng
KW - consecutive primes in tuples; bounded gaps between primes; Maynard-Tao theorem
UR - http://eudml.org/doc/279690
ER -