Problem 386

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claims/: The 2 claim pages of Problem 386, one per claimant's result; the problem's standing derives from them.

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Statement. Let $2\leq k\leq n-2$. Can $\binom{n}{k}$ be the product of consecutive primes infinitely often? For example

$$\binom{21}{2}=2\cdot 3\cdot 5\cdot 7.$$

Status. Open. The site labels the problem OPEN and credits no solution. The standing derives from the claim pages: the two pending partial claims are Martins 2026, a manuscript that bounds the primes and the block length of any sufficiently large solution, proves finiteness for each fixed $k$ and block length, and treats the case $k=2$ in detail, and Mysore 2026, a draft note showing that any further solution with $k=2$ is a product of at least $1924$ consecutive primes; neither decides whether solutions are infinite, so the problem is open with no settling or pending full claim.

Source. erdosproblems.com/386, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #386, https://www.erdosproblems.com/386.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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