Conditional Prime Values for Polynomial-Product Prime Factors

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corollary_4_1_reconstruction: Reconstructs the Bateman--Horn implication that supplies the multiplicative interval hypothesis used in the degree-scale product bound.

evidence/: Source-reading records and independent reconstruction reviews for the conditional polynomial-product argument, without accepted whole-proof credit.

lemma_2_1_reconstruction: Reconstructs the fixed-divisor reduction to an irreducible polynomial with the same degree and no fixed prime divisor.

theorem_3_2_reconstruction: Reconstructs the conditional transfer from prime values in multiplicative intervals to a degree-scale lower bound for the running product.

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Target and status evidence

For an irreducible $f\in\mathbb Z[X]$ of degree $d\geq2$, let

$$F_f(n)=P^+\!\left(\left|\prod_{m=1}^n f(m)\right|\right), \qquad P^+(1)=1.$$

Irreducibility and degree at least two exclude integer roots, so the product is nonzero. The absolute value makes changing the sign of $f$ harmless, as in the source's p. 1 convention.

The target is the degree-scale estimate $F_f(n)\gg_f n^d$ for every such $f$, the stronger question in Problem 976. Neither conditional result below resolves that universal question unconditionally. This lead records the identified note and its remaining obligations, not an exhaustive review of the problem's current status.

Conditional source and exact premise

Aron Bhalla's unpublished five-page note, A Conditional Note on an Erdős Problem on Large Prime Factors of Polynomial Products, is filed as the library source Bhalla (2026). Its card holds the retained PDF, the provenance line identifying those bytes, and the Drive retrieval record. That filing identifies the retained artifact, not publication or mathematical acceptance.

The PDF sat at erdos/research/leads/polynomial_product_prime_value_condition/bhalla_conditional_note.pdf, the path the records under Current review name, until it moved unchanged to the card on 2026-09-17.

Hypothesis 3.1 on physical and numbered p. 3 says that for every irreducible $g\in\mathbb Z[X]$ with positive leading coefficient and no fixed prime divisor, there are constants $A_g>1$ and $X_0(g)$ such that, for every real $X\geq X_0(g)$, some integer $t\in[X,A_gX]$ has $g(t)$ prime.

Under that premise, Theorem 3.2 on p. 3 states that every irreducible $f\in\mathbb Z[X]$ of degree $d\geq2$ satisfies

$$F_f(n)\gg_f n^d$$

for every integer $n\ge N_f$, with $C_f>0$ and $N_f\in\mathbb Z_{\ge1}$ chosen after fixing $f$, so that $F_f(n)\ge C_fn^d$. No uniformity over polynomials is asserted. Corollary 4.1 on p. 5 derives the same conclusion from the Bateman--Horn conjecture for single polynomials, using the positive-input count $1\le t\le X$. The PDF prints only $t\le X$; making the positive-integer domain explicit is a compilation clarification, not an author-issued revision. Both conclusions remain conditional.

The author-recorded reconstruction is split across Lemma 2.1, Theorem 3.2, and Corollary 4.1. It preserves both conditional premises and adds no unconditional, status, or tier claim.

Proposed connection and proof pointer

After replacing $f$ by $-f$ if necessary, assume that its leading coefficient is positive. Lemma 2.1, on physical pp. 2--3, takes the fixed divisor $D=\gcd\{f(m):m\in\mathbb Z\}>0$ and constructs integers $M\ge1$, $0\le a<M$, and an irreducible $h\in\mathbb Z[X]$ of the same degree such that

$$f(a+Mt)=Dh(t),$$

where $h$ has positive leading coefficient and no fixed prime divisor. Fix $D,a,M,h$ before varying $n$, and let $A=A_h>1$ come from Hypothesis 3.1. Theorem 3.2 uses $X_n=\lfloor(n-a)/(AM)\rfloor$. For all sufficiently large integer $n$, take $X_n\ge\max(X_0(h),1)$ and the supplied integer $t\in[X_n,AX_n]$. Then $1\le M\le a+Mt\le a+MAX_n\le n$. Thus the prime $h(t)$ divides an actual factor $f(a+Mt)$ of the prefix product, and its size is bounded below by a positive constant depending on $f$ times $n^d$. The lower endpoint, omitted from the PDF's p. 4 proof, is supplied here. Corollary 4.1 observes that the asserted Bateman--Horn asymptotic supplies a prime value in $(X,2X]$ for every large $X$.

The three linked pages now hold an author-recorded reconstruction of this conditional chain. The existing non-blind reading and grade records remain separate; no independent proof coverage, status, tier, or unconditional E0976 result follows.

Obstacles and next investigation

The prime-value premise is unproved. The note remains unpublished, with no publication or public-acceptance evidence established by this record. The fixed-divisor reduction and transfer to every sufficiently large integer endpoint were checked in the disclosed non-blind source reading below. That scope does not discharge the fresh-context review obligation. The author-recorded reconstruction now makes those three steps of the conditional chain explicit, but the fresh-context review obligation remains.

The disclosed non-blind source reading of Lemma 2.1, Theorem 3.2 and Corollary 4.1 (2026-09-10) — source conventions, integrality, irreducibility, fixed-divisor removal, sign, both endpoints and fixed-polynomial constant dependence — is recorded below, and it is not the fresh-context review this section asks for. Its positive result supports only the conditional implication; neither antecedent is thereby proved. Any attempt to prove Hypothesis 3.1 is new mathematics outside this record.

Current review

The lead's proposed connection needs fresh-context review. The retained source reading checks the full conditional argument against all five PDF pages and exposes its deductions, attacks and checklist. The reading record records the exact subject, visual page coverage, permitted material and pre-existing exposure. The distinct grade is pass with named corrections solely for disclosed non-blind source-only reading, not a tier-bearing whole-claim review.

The reconstruction pages are author work, not a fresh review. They leave review_status: unreviewed and preserve both conjectural premises.

The three reconstruction pages were then independently reviewed and separately graded as a source reconstruction. The independent review and the distinct grade each report faithful with corrections, and the seven merged corrections are applied and mapped in the source-reading record. That review covers the reconstruction of the note's argument against the retained PDF. It does not bear on whether Hypothesis 3.1 or the Bateman--Horn conjecture is true, changes no problem status, and gives the lead itself no accepted whole-proof coverage.

The manuscript is unchanged. Its positive-input convention and omitted lower endpoint are explicitly handled in the retained account. This candidate keeps review_status: unreviewed; it earns no independently accepted whole-proof, status-change, publication, formal-build, native-tier or new problem-solving credit. The prime-value and Bateman--Horn premises remain unproved, and fresh-context review is still required before stronger proof standing is asserted.