Consecutive primes in tuples

• Autorzy: William D. Banks , Tristan Freiberg , Caroline L. Turnage-Butterbaugh
• Języki publikacji: EN

Abstrakty

EN

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 $𝓗(x) = {gx + h_j}_{j=1}^k$ of linear forms in ℤ[x], the set $𝓗(n) = {gn + h_j}_{j=1}^k$ contains at least m primes for infinitely many n ∈ ℕ. In this note, we deduce that $𝓗(n) = {gn + h_j}_{j=1}^k$ 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.

Kategorie tematyczne

• 11N36: Applications of sieve methods
• 11A41: Primes

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2015
• Tom: 167
• Numer: 3
• Strony: 261-266

Daty

wydano

2015

Twórcy

autor

William D. Banks

• Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A.

autor

Tristan Freiberg

• Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A.

autor

Caroline L. Turnage-Butterbaugh

• Department of Mathematics, University of Mississippi, University, MS 38677, U.S.A.

Identyfikatory

DOI

10.4064/aa167-3-4