On the factorization of consecutive integers

A classical result of Sylvester [21] (see also [16], [17]), generalizing Bertrand’s Postulate, states that the greatest prime divisor of a product of k consecutive integers greater than k exceeds k. More recent work in this vein, well surveyed in [18], has focussed on sharpening Sylvester’s theorem, or upon providing lower bounds for the number of prime divisors of such a product. As noted in [18], a basic technique in these arguments is to make a careful distinction between “small” and “large” primes, and then apply sophisticated results from multiplicative number theory. Along these lines, if we write ( n k ) = U · V, n ≥ 2k,