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Claim. Let $f\in\mathbb Z[x]$ be irreducible of degree $d\geq2$ and let $F_f(n)$ be the greatest prime factor of $\prod_{m\le n}f(m)$, as in Problem 976. Theorem 3.2 of Aron Bhalla's five-page note A Conditional Note on an Erdős Problem on Large Prime Factors of Polynomial Products asserts that

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

for all sufficiently large $n$, the stronger of the problem's two questions, under the note's Hypothesis 3.1: every irreducible $g\in\mathbb Z[x]$ with positive leading coefficient and no fixed prime divisor has constants $A_g>1$ and $X_0(g)$ such that each real $X\geq X_0(g)$ admits an integer $t\in[X,A_gX]$ with $g(t)$ prime. The argument first removes the fixed divisor $D$ of $f$ (Lemma 2.1): after making the leading coefficient positive, there are integers $M\geq1$ and $0\leq a<M$ and an irreducible $h$ of degree $d$ with $f(a+Mx)=Dh(x)$, where $h$ has no fixed prime divisor. The hypothesis, applied to $h$ at the scale $X_n=\lfloor(n-a)/(A_hM)\rfloor$, supplies a prime $h(t)$ with $1\le a+Mt\le n$; that prime divides the factor $f(a+Mt)$ of the running product and has size of order $n^d$. Corollary 4.1 derives Hypothesis 3.1, with $A_g=2$, from the Bateman--Horn asymptotic $\pi_g(X)\sim c_gX/\log X$ for single polynomials, so the same bound follows from Bateman--Horn. The note claims no unconditional result, and the implied constant depends on $f$. The note is digested on its card; the research lead Conditional Prime Values for Polynomial-Product Prime Factors holds an author-recorded reconstruction of the conditional chain, which supplies the lower endpoint $1\le a+Mt$ that the note's proof omits.

Hypotheses. Hypothesis 3.1 is unproved, and so is the Bateman--Horn conjecture that implies it. Neither is known for any irreducible polynomial of degree at least two: no polynomial of degree at least two is known to take infinitely many prime values. The claim therefore settles neither question of the problem unconditionally.

Depends on. No page of this wiki. The claim rests on the cited note alone; the reconstruction pages of the lead named above are research records, not premises.

Standing. An unpublished note shared from a Google Drive address and announced in the site's discussion thread on 16 April 2026; the announcing post says the note was produced using GPT 5.4. The PDF prints no date or version, and its metadata gives 16 April 2026. It has no arXiv record, no refereeing and no formalization. A reply in the thread the same day reports a check of the argument that found no issue; it is not acceptance evidence. On 2026-10-05 the site labeled the problem OPEN (page last edited 1 February 2026) with no proof claim, and the community's AI-contributions wiki listed the note as a conditional partial result. The lead named above records a non-blind reading of the conditional argument against the PDF and an independent review of its reconstruction; both concern the implication only and bear neither on the hypothesis nor on the problem's standing.