[#P21] Bunyakovsky conjecture

Problem. Let \(f\in\mathbb{Z}[x]\) be irreducible, with positive leading coefficient and no fixed prime divisor. Then \(f(n)\) is prime for infinitely many positive integers \(n\).

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Definitions and notation

1Context

The conjecture extends Dirichlet's theorem on primes in arithmetic progressions to one-variable polynomials of arbitrary degree.

2Problem setup

Definition 1 (A fixed prime divisor). A fixed prime divisor is a prime that divides f(n) for every integer n.

Remark 1. Irreducibility is over the integers, equivalently over the rational numbers for primitive polynomials.

3What counts as a solution

Prove infinitude of prime values for every polynomial satisfying the hypotheses, or give such a polynomial and prove that it takes prime values only finitely often.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: The linear case follows from Dirichlet's theorem. The linked AMS publication presents the irreducible higher-degree prime-values statement as open; Crossref proof claims were not confirmed by an established source. Exact unresolved remainder: Prove infinitely many prime values for every irreducible integer polynomial satisfying the stated positivity and fixed-divisor hypotheses, or give a qualifying polynomial with only finitely many prime values.^[2][1]

1Packet records

Show recorded summaries

Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: The linear case follows from Dirichlet's theorem. The linked AMS publication presents the irreducible higher-degree prime-values statement as open; Crossref proof claims were not confirmed by an established source. Exact unresolved remainder: Prove infinitely many prime values for every irreducible integer polynomial satisfying the stated positivity and fixed-divisor hypotheses, or give a qualifying polynomial with only finitely many prime values.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: The linear case follows from Dirichlet's theorem. The linked AMS publication presents the irreducible higher-degree prime-values statement as open; Crossref proof claims were not confirmed by an established source. Exact unresolved remainder: Prove infinitely many prime values for every irreducible integer polynomial satisfying the stated positivity and fixed-divisor hypotheses, or give a qualifying polynomial with only finitely many prime values.^[2][1] Relevance to this problem For Bunyakovsky conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 2, Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: The linear case follows from Dirichlet's theorem. The linked AMS publication presents the irreducible higher-degree prime-values statement as open; Crossref proof claims were not confirmed by an established source. Exact unresolved remainder: Prove infinitely many prime values for every irreducible integer polynomial satisfying the stated positivity and fixed-divisor hypotheses, or give a qualifying polynomial with only finitely many prime values. Open full record →
Prove infinitude of prime values for every polynomial satisfying the hypotheses, or give such a polynomial and prove that it takes prime values only finitely often.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove infinitude of prime values for every polynomial satisfying the hypotheses, or give such a polynomial and prove that it takes prime values only finitely often.^[1] Relevance to this problem For Bunyakovsky conjecture, record bunyakovsky-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove infinitude of prime values for every polynomial satisfying the hypotheses, or give such a polynomial and prove that it takes prime values only finitely often. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove infinitude of prime values for every polynomial satisfying the hypotheses, or give such a polynomial and prove that it takes prime values only finitely often. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

Original intake status. The cited AMS publication presents Bunyakovsky's assertion as an open conjecture. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

Dirichlet's theorem proves the degree-one case. No single irreducible polynomial of degree greater than one is known to take prime values infinitely often.

Recorded example 1. The polynomial n^2 + 1 satisfies the hypotheses, and its prime values are conjectured to be infinite.

Computational notes

Testing polynomial values through any finite input range cannot establish infinitude.

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Bunyakovsky conjecture.” TheoremDB. P21. Problem statement; statement text SHA-256 4c4275943f3c55ef46d5171a662dfef2ffc34e0d63aff112934a2e87b2825f11. https://theoremdb.org/statement/?ref=P21

BibTeX

@misc{theoremdb-problem-4c4275943f3c55ef46d5171a662dfef2ffc34e0d63aff112934a2e87b2825f11,
  title = {{Bunyakovsky conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 4c4275943f3c55ef46d5171a662dfef2ffc34e0d63aff112934a2e87b2825f11},
  url = {https://theoremdb.org/statement/?ref=P21}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from ams.org^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Jean-Marie De Koninck and Nicolas Doyon, The Life of Primes in 37 Episodes, American Mathematical Society, 2021, episode 37.1.2. American Mathematical Society book preview, episode 37.1.2, Bunyakovsky conjecture. ↗website · primary source · checked 2026-07-31Source use: original summary.The cited AMS publication presents Bunyakovsky's assertion as an open conjecture. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at episode 37.1.2, Bunyakovsky conjecture.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.States the prime-values conjecture and distinguishes it from known linear cases.Source named by the research packet.
P. G. L. Dirichlet, Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält, Abhandlungen der Königlich Preussischen Akademie der Wissenschaften zu Berlin (1837), 45-81; reprinted in G. Lejeune Dirichlet's Werke I (1889), 313-342. DOI 10.3931/e-rara-17606. collected works volume I, page 341, conclusion for primes in an admissible arithmetic progression. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Proves Bunyakovsky's prediction for every degree-one polynomial under the admissibility hypotheses and supplies no higher-degree case.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P21] Bunyakovsky conjecture

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3What counts as a solution

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Computational notes

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Recent contributions

Computational notes

2See also

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Background

Definitions

What counts as a solution

What's known

Formal statement

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Related work

Record & provenance