Statement copy in the gpt 6 astra pro collection (Math-140)

Bunyakovsky Conjecture

Definitions and mathematical statement

For a nonconstant integer polynomial $f$, a fixed prime divisor is a prime $p$ dividing $f(n)$ for every integer $n$. Suppose $f$ is irreducible over ${\mathbb{Q}}$, primitive, and has positive leading coefficient, with \[ \forall p\in\mathcal P\ \exists n\in{\mathbb{Z}}:\ p\nmid f(n). \] The assertion is that the set $\{n\in{\mathbb{Z}}_{>0}:f(n)\in\mathcal P\}$ is infinite. Positive leading coefficient ensures that signs cannot obstruct eventual positive prime values. No explicit count or density is required.

Formulation and concept references: [S1].

Short English statement

Must every irreducible positive-leading integer polynomial without a fixed prime divisor produce infinitely many primes?

Sources

• S1 Eric W. Weisstein with Ed Pegg Jr., Bouniakowsky Conjecture, MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BouniakowskyConjecture.html. Catalogue formulation source.
• E1 R. J. Lemke Oliver, Almost-primes represented by quadratic polynomials, Acta Arithmetica 151 $2012$, no. 3, 241--261. Theorem 1 (squarefree $P_2$ values for admissible quadratics), and Section 2.2, Lemma 2 (the linear-sieve functions and bilinear remainder). https://lemkeoliver.github.io/papers/04-Quadratic.pdf. Author's manuscript consulted 25 September 2026.
• E2 T. Browning, E. Sofos, and J. Teräväinen, Bateman--Horn, polynomial Chowla and the Hasse principle with probability 1, arXiv:2212.10373v2, 21 May 2026. Theorems 1.2 and 1.5, including the coefficient-box and exceptional-set quantifiers. https://arxiv.org/html/2212.10373v2. Consulted 25 September 2026; results averaged over polynomials are distinguished from an assertion for every fixed polynomial.
• E3 L. Grimmelt and J. Merikoski, On the greatest prime factor and uniform equidistribution of quadratic polynomials, arXiv:2505.00493 $2025$, abstract and introduction, the $n^{1.312}$ greatest-prime-factor bound for $n^2+1$. https://arxiv.org/abs/2505.00493. Consulted 25 September 2026; a greatest-prime-factor result, not a prime-values theorem.