Consecutive primes in tuples

Abstract

In a recent 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 Erd\H os and Turán by producing strings of $m+1$ consecutive primes whose successive gaps ${\delta }_1,\dots ,{\delta }_m$ form an increasing (resp. decreasing) sequence. We also show that such strings exist with ${\delta }_{j-1}\mid {\delta }_j$ 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 $a\mathrm{mod}D$.

Revised version

References in corpus (2)

• New equidistribution estimates of Zhang type
• Primes in intervals of bounded length

Cited by in corpus (6)

• Bounded gaps between primes with a given primitive root
• The "bounded gaps between primes" Polymath project - a retrospective
• Arithmetic functions at consecutive shifted primes
• Primes in intervals of bounded length
• Variants of the Selberg sieve, and bounded intervals containing many primes
• Configurations Of Consecutive Primitive Roots