Problem 383

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claims/: The 0 claim pages of Problem 383, one per claimant's result; the problem's standing derives from them.

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Statement. Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime divisor of

$$\prod_{0\leq i\leq k}(p^2+i)$$

is $p$?

Status. Open, the site's label (OPEN; the site marks the problem as not decidable by a finite computation). Its proof-claims tab carries one partial proof claim, submitted 2026-07-25 and without response which settles no case of the question and gets no claim page; the Current assessment records it with the reason.

Source. erdosproblems.com/383, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #383, https://www.erdosproblems.com/383.

Formalization. Statement in formal-conjectures.

Current assessment

Open; no result decides a single case. The site formulation above asks, for every $k$, for infinitely many primes $p$ such that $p$ is the largest prime factor of $\prod_{0\le i\le k}(p^2+i)$, that is, every $p^2+i$ with $1\le i\le k$ has all its prime factors below $p$. The site's commentary notes that a positive answer would answer the second part of Problem 382, and that the heuristic probability $1-\log2$ that an integer $n$ has no prime factor of size at least $n^{1/2}$ predicts the answer yes. No source proves the statement for any $k\ge1$, and the site labels the problem OPEN.

Proof claim without a page. The site's proof-claims tab lists a partial proof claim by Rafik Zeraoulia (using OpenAI GPT-5.6 Thinking, as the tab names the system), submitted 2026-07-25, with the write-up A positive-proportion smoothness bound and computational results for Erdős Problem 383 (Zenodo record 21544693, 2026-07-25, CC BY 4.0). It asserts that for every fixed $k\ge1$ and $\varepsilon>0$ a positive proportion of primes $p$ satisfy $P^+(\prod_{i=0}^k(p^2+i))\le p^{2-1/(2k)+\varepsilon}$, by applying a theorem of Dartyge, Martin and Tenenbaum on smooth values of polynomials to $\prod_{i=1}^k(X^2+i)$ and restricting to primes near $x$, and it reports an exhaustive computation over the primes $p\le10^8$: the prime $p=93\,609\,881$ is the largest prime factor of the product for $k=12$, the only such prime below $10^8$, and no prime in that range works for $k=13$. The claim gets no page because it settles no instance of the problem: the exponent $2-1/(2k)+\varepsilon$ is at least $3/2$ for every $k\ge1$, where the question needs exponent $1$, and the computation exhibits single primes, not infinitely many; the write-up itself says that the bound does not resolve the exponent-$1$ problem and that the computation proves no infinitude.

Forum remarks. In the problem's discussion thread (two comments as of 2026-10-07), a comment of 2026-06-21 reports that the prime $p=9\,188\,057$ is the largest prime factor of the product for every $k\le10$ and fails at $k=11$, where $p^2+11$ has the prime factor $1\,407\,006\,523\,921>p$, and gives the counts of primes $p\le10^7$ satisfying the condition through each $k$ ($181\,281$ for $k=1$ down to one for $k=10$ and none for $11\le k\le15$); the commenter discloses that the computation and comment were prepared with assistance from Codex 5.5 and ChatGPT 5.5 Pro. A comment of 2025-09-08 observes that the first nontrivial case, $k=1$, would follow from the conjecture that there are infinitely many Newman–Shanks–Williams primes, and asks for an unconditional proof; none is recorded. Neither remark settles an instance.

Formalization and search scope. The formal-conjectures statement file ErdosProblems/383.lean, as of its last change on 18 September 2026, states erdos_383 as answer(sorry) ↔ ∀ k, {p : ℕ | p.Prime ∧ Nat.maxPrimeFac (∏ i ∈ Finset.Icc 0 k, (p ^ 2 + i)) = p}.Infinite, category research open, with no formal_proof attribute. Search scope: the site's page, discussion thread and proof-claims tab, the Zenodo record of the proof claim, the formal-conjectures file and its commit history, and the community database, which lists the problem as open and formalized as of its last update on 2025-08-31, without dating when either state was set.