Let L = L_1,…,L_K be a set of distinct linear forms L_k(n) = a_kn+b_k, where the coefficients a_k,b_k are positive integers. For a prime p, let omega_L(p) denote the number of roots of ∏_k=1^K L_k(n) modulo p. The set L is admissible if omega_L(p) < p for every prime p. Prime K-tuples conjecture. If L is admissible, then there are infinitely many integers n such that L_1(n),…,L_K(n) are all prime. This is the prime-tuples conjecture in the linear-forms setting and would provide simultaneous primality for every admissible finite family. It is used here as the qualitative hypothesis underlying the paper's conditional results; no resolution is given in the supplied text.

Prime $K$ -tuples conjecture

Let $\mathcal{L} = \{L_1,\ldots,L_K\}$ be a set of distinct linear forms $L_k(n) = a_kn+b_k$ , where the coefficients $a_k,b_k$ are positive integers. For a prime $p$ , let $\omega_\mathcal{L}(p)$ denote the number of roots of $\prod_{k=1}^K L_k(n)$ modulo $p$ . The set $\mathcal{L}$ is admissible if $\omega_\mathcal{L}(p) < p$ for every prime $p$ . Prime $K$ -tuples conjecture. If $\mathcal{L}$ is admissible, then there are infinitely many integers $n$ such that $L_1(n),\ldots,L_K(n)$ are all prime.

This is the prime-tuples conjecture in the linear-forms setting and would provide simultaneous primality for every admissible finite family. It is used here as the qualitative hypothesis underlying the paper's conditional results; no resolution is given in the supplied text.

Equivalent formulations 2

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The prime $k$ -tuples conjecture
Let $b_1,\ldots,b_k$ be distinct integers. The tuple is admissible if, for every prime $p$ , there is a congruence class modulo $p$ containing none of the $b_i$ . The prime $k$ -tuples conjecture. If $b_1,\ldots,b_k$ is an admissible $k$ -tuple of integers, then there exist infinitely many positive integers $n$ such that
$n+b_1,\ldots,n+b_k$
are all prime.

This is a central conjecture about simultaneous prime values of admissible linear patterns and remains open in general.
source: Christian Axler, “Some Results on a Conjecture of Hardy and Littlewood”, arXiv:1909.12625 (2019).
The prime $k$ -tuples conjecture
Let $\mathcal{H}=\{h_1,\ldots,h_k\}$ be a set of distinct integers. It is admissible if $\#\{\mathcal{H}\,\, (\operatorname{mod}\,p)\}<p$ for every prime $p$ . The prime $k$ -tuples conjecture. If $\mathcal{H}$ is admissible, then there exist infinitely many integers $n$ such that the translates $n+h_1,\ldots,n+h_k$ are prime. This is an outstanding problem in analytic number theory and is far beyond current techniques; sieve methods establish only weaker simultaneous-primality results.
source: Oliver McGrath, “A variation of the prime k-tuples conjecture with applications to quantum limits”, arXiv:2008.11119 (2022).

Sources & referencesView supporting material

Primary source

Kyle Pratt, “The irrationality of a prime factor series under a prime tuples conjecture”, arXiv:2409.15185 (2024).

Additional references

9 papers in this index state this conjecture (2012–2024). The statement above is taken from the most recent of them; the others are arXiv:2402.00748, arXiv:2301.05044, arXiv:2111.00608, arXiv:1910.13450, arXiv:1910.14674, arXiv:1407.1747, arXiv:1311.4600, arXiv:1205.4610.

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Prime $K$ -tuples conjecture

Equivalent formulations 2

The prime $k$ -tuples conjecture

The prime $k$ -tuples conjecture

Primary source

Additional references

Progress summary

Solutions 0

Equivalent formulations 2

The prime kkk-tuples conjecture

The prime kkk-tuples conjecture

Primary source

Additional references

Progress summary

The prime $k$ -tuples conjecture

The prime $k$ -tuples conjecture