[#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 \(n\).
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.
Definition 2 (Irreducibility). Irreducibility is over the integers, equivalently over the rational numbers for primitive polynomials.
Remark 1. The conjecture extends Dirichlet's theorem on primes in arithmetic progressions to one-variable polynomials of arbitrary degree.
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
Current 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
Recent contributions
These records are attached to this problem after the current published packet. Each badge shows its current verification or packet-review step.
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
How to cite
TheoremDB contributors, “Bunyakovsky conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/bunyakovsky-conjectureThis page as plain text: bunyakovsky-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from ams.org[1], current as of July 31, 2026.
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.