The board / number theory / #386
Erdős Problem #386
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(2 \le k \le n - 2\). Can \(\binom{n}{k}\) be the product of consecutive primes infinitely often? Here \(k\) may vary with \(n\): the question asks for infinitely many admissible binomial coefficients, not for a single \(k\) that works infinitely often.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/386.Prizenone
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA280992
Original pageerdosproblems.com/386
Attempts
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Proved an elementary prime-gap reduction and, modulo Granville–Ramaré plus PNT, isolated all sufficiently large solutions to a thin prime-free edge regime; exact certified searches found only the nine
PARTIAL