# Consecutive primes in tuples

Published

Journal Article

© Instytut Matematyczny PAN, 2015. 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 script H(x) = {gx + hj}j=1k of linear forms in double-struck Z[x,] the set script H(n) = {gn + hj}j=1k contains at least m primes for infinitely many n ∈ double-struck N. In this note, we deduce that script H(n) = (gn + hj}j=1k contains at least m consecutive primes for infinitely many n ∈ double-struck N. We answer an old question of Erdo{combining double acute accent}s and Turán by producing strings of m + 1 consecutive primes whose successive gaps δ1,...,δm form an increasing (resp. decreasing) sequence. We also show that such strings exist with δj-1 | δj 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.

### Full Text

### Cited Authors

- Banks, WD; Freiberg, T; Turnage-Butterbaugh, CL

### Published Date

- January 1, 2015

### Published In

### Volume / Issue

- 167 / 3

### Start / End Page

- 261 - 266

### International Standard Serial Number (ISSN)

- 0065-1036

### Digital Object Identifier (DOI)

- 10.4064/aa167-3-4

### Citation Source

- Scopus