On a theorem of Sylvester and Schur

A classical theorem discovered independently by J. Sylvester [8], and I. Schur [7], states that the product of k consecutive integers, each greater than k, has a prime divisor greater than k. The most elementary proof of this theorem is due to P. Erdos [3]. In the present paper we will prove a recent conjecture of P. Erdos and obtain as a corollary a " best possible " refinement of the Sylvester-Schur theorem announced by L. Moser [5]. THEOREM. Let pk be the least prime ^ 2k. If n^pk then { , | has a prime divisor>pk, with the exceptions I I, I I. The example 6-7-8-9-10 shows that the following refinement of the Sylvester-Schur theorem is best possible: COROLLARY. If n ^ 2k then I, I has a prime divisor ^\k. Throughout the paper p denotes a prime and p*\\m indicates that pa\m but pa+xlm. Recently J. Rosser and L. Schoenfeld [6], obtained fairly precise estimates for 6(x) = 2 log ̂> and TT{X) = 2 1: 0{x)<l-01624a; for 0<x, (1) 7r(rc)<l-25506x/loga; for l<x. (2) Using these results we are able to establish the theorem by methods which are, to some extent, modelled on a paper by P. Erdos [3]. Proof of the theorem. Assume I, I has no prime divisors ^ 2k. Since I, I implies pa ^ n, (see paper of P. Erdos [3]), | ( 8 p<2fc From (2) it follows that < w < 7 l fo