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Dickson's conjecture

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Let k≥1k \ge 1 and let a1,…,aka_1, \dots, a_k and b1,…,bkb_1, \dots, b_k be integers with bi≥1b_i \ge 1 for every ii, and consider the kk linear forms

a1+b1n,a2+b2n,…,ak+bkn.a_1 + b_1 n, \quad a_2 + b_2 n, \quad \dots, \quad a_k + b_k n .

Suppose that no prime divides all the values of the product, that is, suppose that for every prime pp there exists an integer nn with

p∤∏i=1k(ai+bin).p \nmid \prod_{i=1}^{k} (a_i + b_i n).

Then there are infinitely many positive integers nn such that the kk numbers a1+b1n,…,ak+bkna_1 + b_1 n, \dots, a_k + b_k n are all prime.

Equivalent formulations 1Other wordings

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

  1. Dickson's prime kk-tuples conjecture

    Let k≥1k\ge1, and let A1,…,Ak,B1,…,BkA_1,\dots,A_k,B_1,\dots,B_k be integers with Aj>0A_j>0 for j=1,…,kj=1,\dots,k. Suppose that for every prime pp there exists an integer npn_p such that

    p∤∏j=1k(Ajnp+Bj).p\nmid\prod_{j=1}^k(A_jn_p+B_j).

    Dickson's prime kk-tuples conjecture. There exist infinitely many integers nn such that Ajn+BjA_jn+B_j is prime for 1≤j≤k1\le j\le k. This conjecture, denoted TC(k)\mathrm{TC}(k) in the paper, is attributed to Dickson (1904) and generalizes the assertion that admissible linear forms simultaneously take prime values; it remains open except in limited cases.

    source: Pieter Moree, “Primitive root producing quadratics”, arXiv:math/0406033 (2004).

References

Primary source

Wikipedia

Additional references

  1. Wikipedia, Dickson's conjecture, the article this problem comes from.
number-theoryprime-numbersunsolved-problem
Posted

Progress summary

Refreshed
Claimed progress

Dickson’s conjecture remains unproved, with a new unrefereed manuscript offering computational certificates but no verified resolution.

Leonard Dickson proposed the conjecture in 1904: any finite admissible family of linear forms should be simultaneously prime infinitely often. The case k=1k=1 is Dirichlet’s theorem, while twin primes and other prime-pattern conjectures are special cases.

Known results

  • Dirichlet’s theorem proves the one-form case k=1k=1.
  • Zhang (2013) proved a weaker DHL(k,2)\mathrm{DHL}(k,2) statement for sufficiently large kk, yielding bounded prime gaps.
  • Maynard–Tao proved that sufficiently large admissible families contain infinitely many inputs where at least mm forms are prime, for every mm; this does not give simultaneous primality of all forms.

September 3, 2026 certificate manuscript

On September 3, 2026, Ivan Anisenya’s Zenodo manuscript proposed explicit computational certificates and a framework relating Dickson’s conjecture to prime-constellation counts. It does not establish the conjecture; the deposit is unrefereed and its certificates remain unverified.

Current status (as of September 2026): Dickson’s conjecture remains open; classical partial results and the new certificate claims do not prove simultaneous primality of every admissible family.

Sources

Solutions 0

No solutions have been posted yet.