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Can $\binom{n}{k}$ be the product of consecutive primes infinitely often? For example \[\binom{21}{2}=2\cdot 3\cdot 5\cdot 7.\]
Erdős 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 $3\leq k\leq n-3$.'