Problem 386

Erdős Problem #386: declared status 'open'. Formalized: yes. Let $2\leq k\leq n-2$. Can $\binom{n}{k}$ be the product of consecutive primes infinitely often? For example$$\binom{21}{2}=2\cdot 3\cdot 5\cdot 7.$$ Current best: Erd\H{o}s and Graham write that 'a proof that this cannot happen infinitely often for $\binom{n}{2}$ seems hopeless; probably this can never happen for $\binom{n}{k}$ if $3\leq k\leq n-3$.' Weisenberg has provided four easy examples that show Erd\H{o}s and Graham were too optimistic here:$$\binom{7}{3}=5\cdot 7,$$$$\binom{10}{4}= 2\cdot 3\cdot 5\cdot 7,$$$$\binom{14}{4} = 7\cdot 11\cdot 13,$$and$$\binom{15}{6}=5\cdot 7\cdot 11\cdot 13.$$The known values of $n$ for which $\binom{n}{2}$ is the product of consecutive primes are $4,6,15,21,715$ (see A280992). Prize: no. OEIS: A280992. Tags: binomial coefficients, number theory.