Hardy-Littlewood Conjecture A (Prime k-tuples)

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $k\ge1$ and let $a_1,\ldots,a_k\in\mathbb Z$ be fixed offsets; repeated offsets may be deleted without changing the prime conditions. The set of offsets is admissible if, for every prime $p$, there is some residue $m\pmod p$ for which \[ \gcd(m+a_i,p)=1\qquad(1\le i\le k). \] Equivalently, the residues $-a_i\pmod p$ do not cover all residues modulo $p$. A prime is a positive integer larger than one with exactly two positive divisors.

Statement recorded in the supplied source.

For every fixed admissible collection of integer offsets $a_1,\ldots,a_k$, are there infinitely many $n\in\mathbb N$ such that \[ n+a_1,\ldots,n+a_k \] are all prime? Equivalently, does such an $n$ exist above every prescribed bound? [S2]

Short English statement

Does every fixed prime pattern without a modular obstruction occur infinitely often among translates of the integers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1517 (HL-A). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 James Maynard, Small gaps between primes, Annals of Mathematics 181 (2015), 383--413; arXiv:1311.4600, Section 1, admissibility and the qualitative Prime $k$-tuples Conjecture. https://arxiv.org/abs/1311.4600.