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Consecutive primes in tuples

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Banks W., Freiberg T., Turnage-Butterbaugh C.

Acta Arithmetica

2015

tom 167

nr 3

261-266

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.

Ograniczanie wyników

1 Acta Arithmetica

1 Banks W.

1 Freiberg T.

1 Turnage-Butterbaugh C.

1 2015

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Consecutive primes in tuples