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Claim. Pedro Martins's manuscript "Consecutive-prime products among binomial coefficients" (Zenodo record 23037725, doi:10.5281/zenodo.23037725, published 2026-09-29 under CC BY 4.0) studies the solutions $(n,k)$ with $2\le k\le n-2$ of $\binom nk=p_ap_{a+1}\cdots p_b$, the set whose finiteness Problem 386 asks about. Write $p_a\le p_b$ for the end primes of a solution and $L=b-a+1$ for its block length, and take $k\le n/2$ by the symmetry $k\mapsto n-k$. Its Theorem A proves that, apart from finitely many solutions, $p_b\le n/2$, no prime lies in $(n-k,n]$, and $k<\exp(\tau_1(\log n)^{2/3}(\log\log n)^{1/3})$ for an absolute $\tau_1>0$, a bound the manuscript quotes as Theorem 2 of Granville and Ramaré (Mathematika, 1996), which applies because every solution is squarefree; that every sufficiently large solution has $p_b\ll n^{25/37}$, $p_a,p_b\ll_\varepsilon(k\log n)^{25/12+\varepsilon}$ for every $\varepsilon>0$, and $L\gg k(\log n/\log\log n)^{1/3}$; that the number of solutions with $n\le X$ is at most $\exp(O((\log X)^{2/3}(\log\log X)^{1/3}))=X^{o(1)}$; and that, up to the symmetry, the solutions supported on at most three primes are $(4,2)$, $(6,2)$, $(7,3)$, $(14,4)$ and $(15,2)$. Theorem 10.1 proves that for each fixed $k\ge2$ and each fixed block length $\ell$ there are only finitely many solutions, and Remark 10.2 says that this does not bound the whole set, since an infinite family could have $k$ or $\ell$ unbounded. Proposition 8.1 settles the central diagonal: with $n=2k$ the only solution is $(4,2)$, since Granville and Ramaré proved $\binom{2k}{k}$ not squarefree for $k>4$ and the cases $k\le4$ are checked. For $k=2$ the branches $p_a\in\{3,2\}$ are the equations $p_b\#=n(n-1)$ and $2p_b\#=n(n-1)$, with $17\#=714\cdot715$ the largest known instance; Theorem D shows that their admissible end primes $p_b$ have density zero among the primes, by an averaged Chebotarev density theorem of Lemke Oliver and Smith, that under the generalized Riemann hypothesis there are $O(X^{1/2}(\log X)^4)$ of them up to $X$, and that below $10^{11}$ they are $\{2,3,5,7,17\}$ and $\{3,7\}$; Proposition 5.1 checks that $\binom n2$ is a product of consecutive primes for $4\le n\le5\cdot10^7$ only at $n\in\{4,6,15,21,715\}$. Theorem B gives finiteness under hypotheses: a lower bound on signed sums of the logarithms of a block of consecutive primes gives finiteness for $k=2$, a smoothness hypothesis on $n(n-1)(n-2)$ gives it for each fixed $k\ge3$, and, together with the first hypothesis, a smooth-tuple hypothesis on runs of consecutive integers gives finiteness of the whole set. Theorem E settles the analogue over $\mathbb F_q[T]$ completely.

Submission note. Posted to erdosproblems.com as a proof claim by Pedro Martins (account PedroMartins) on 29 September 2026:

This article studies representations of binomial coefficients as products of consecutive primes. It develops structural and asymptotic restrictions on possible solutions using p-adic valuations, squarefreeness properties of binomial coefficients, estimates for primes in short intervals, and Diophantine methods. Particular attention is given to the case k = 2, which reduces to primorial equations involving consecutive integers and captures a central obstruction to a general finiteness result. The work also investigates bounds on the prime endpoints and block length, sparsity of admissible solutions, conditional finiteness criteria, and computational verification over a large explicit range.

Covers. The restrictions above on every sufficiently large solution, apart from the bound on $k$, which is Granville and Ramaré's; the classification of the solutions on at most three primes; finiteness for each fixed pair $(k,\ell)$; and the central diagonal $n=2k$, where only $(4,2)$ occurs. Each of these is a partial no: it leaves only finitely many solutions of a given shape without deciding the question. The counting estimate and the density-zero and computational statements for $k=2$ are not covered: the manuscript says that Theorem D leaves room for an infinite but very sparse set of endpoints, and a count or a search below a bound excludes no infinite family. Theorem B is conditional on the unproved hypotheses stated above (the lower bound on signed sums of logarithms of consecutive primes, the smoothness hypothesis on $n(n-1)(n-2)$ and the smooth-tuple hypothesis) and settles no instance, so it is not covered. The manuscript says that no argument in it proves finiteness even for $k=2$, and that its results narrow the form of a possible infinite family without deciding whether one exists.

Depends on. No page of this wiki.

Standing. Posted on the problem's proof-claims tab as a partial claim on 2026-09-29 with the Zenodo record as its external link; the entry had no comments, and the claim declares no AI assistance. The manuscript is not refereed and no outside reviewer has recorded accepting it, so the claim is claimed. The site labels the problem OPEN and its remarks do not mention the manuscript.