# P21: Bunyakovsky conjecture

- ID: `P21`
- Reference: `bunyakovsky-conjecture`
- Page: https://theoremdb.org/statements/P21
- Record maturity: Reviewed problem with recorded work

## 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 \(n\).

### Context

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

### Problem setup

- **Definition (A fixed prime divisor).** A fixed prime divisor is a prime that divides f(n) for every integer n.
- **Definition (Irreducibility).** Irreducibility is over the integers, equivalently over the rational numbers for primitive polynomials.
- **Remark.** The conjecture extends Dirichlet's theorem on primes in arithmetic progressions to one-variable polynomials of arbitrary degree.

### What 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.

## Status

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](#reference-2) [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- Original intake status: The cited AMS publication presents Bunyakovsky's assertion as an open conjecture. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The formulation and status were checked against the cited AMS publication on 2026-07-22.
- 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: The polynomial n^2 + 1 satisfies the hypotheses, and its prime values are conjectured to be infinite.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Computational notes

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

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `bunyakovsky-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://www.ams.org/bookstore/pspdf/mbk-139-prev.pdf
   - Also cited at episode 37.1.2, Bunyakovsky conjecture
   - Also cited at Editorial research route recorded 2026-07-31
   - website; primary source; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>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 https://www.e-rara.ch/download/pdf/5688045.pdf
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Proves Bunyakovsky's prediction for every degree-one polynomial under the admissibility hypotheses and supplies no higher-degree case.