Statement copy in the gpt 6 astra pro collection (Math-61)

Schinzel's Hypothesis H

Definitions and mathematical statement

Let $f_1,\ldots,f_r\in{\mathbb{Z}}[x]$ be nonconstant irreducible polynomials with positive leading coefficients. The local admissibility condition is \[ \forall p\text{ prime}\ \exists n\in{\mathbb{Z}}:\quad p\nmid\prod_{i=1}^r f_i(n). \] The proposed conclusion is that $\{n\ge1:f_i(n)\text{ is prime for every }i\}$ is infinite. Irreducibility is over ${\mathbb{Q}}$ for primitive integer polynomials; local admissibility excludes fixed prime divisors. No asymptotic frequency is included in this qualitative assertion.

Formulation and concept references: [S1].

Short English statement

If irreducible integer polynomials have no fixed local obstruction, must they simultaneously take prime values infinitely often?

Sources

• S1 Federico Scavia, Arithmetic Geometry and Algebraic Groups, Conjecture 3.1 $2026$. https://www.lms.ac.uk/sites/default/files/inline-files/Scavia%20-%20Arithmetic%20Geometry%20and%20Algebraic%20Groups.pdf. Catalogue formulation source.
• S2 On The Schinzel–Wójcik Problem Under Hypothesis H. https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/on-the-schinzelwojcik-problem-under-hypothesis-h/C75A7C818639ADA780DE2A47C8BCFA19. Catalogue research/status reference; not a proof endorsement.
• S3 From Golomb to Schinzel. https://www.preprints.org/manuscript/202505.0651. Catalogue research/status reference; not a proof endorsement.
• E1 Alexei N. Skorobogatov and Efthymios Sofos, Schinzel Hypothesis on average and rational points, Inventiones Mathematicae 231 $2023$, 673--739; arXiv:2005.02998; DOI 10.1007/s00222-022-01153-6. Introduction and Theorems 1.1--1.2. https://arxiv.org/abs/2005.02998. Added research source: Fixed-family formulation and coefficient-averaged existence of a simultaneous prime value; the quantifier limitation is retained. Consulted 24 September 2026.
• E2 Tim Browning, Efthymios Sofos and Joni Teräväinen, Bateman--Horn, polynomial Chowla and the Hasse principle with probability 1, arXiv:2212.10373v2, 21 May 2026, Theorems 1.2 and 1.4 and the introduction. https://arxiv.org/abs/2212.10373v2. Added research source: Averaged asymptotics and logarithmic exceptional-proportion savings in coefficient boxes; no result along every fixed affine orbit is assumed. Consulted 24 September 2026.
• E3 John Friedlander and Henryk Iwaniec, Asymptotic sieve for primes, Annals of Mathematics 148 $1998$, 1041--1065; arXiv:math/9811186, Section 1. https://arxiv.org/abs/math/9811186. Added research source: Parity obstruction and the need for additional analytic information beyond ordinary congruence counts. Consulted 24 September 2026.
• E4 Huan Xiao, From Golomb to Schinzel, Preprints.org manuscript 202505.0651, version 2, posted 9 June 2025, Section 3, especially the transition from $3.8$ to $3.9$ and the paragraph following $3.14$. https://www.preprints.org/manuscript/202505.0651. Added research source: Claimed solution, not endorsed. The attempt gives a coefficient-level counterexample to the displayed identity and an independent test of the limiting justification. Consulted 24 September 2026.