# Are there any theorems about a prime $p > k$ in a sequence stronger than Sylvester-Schur?

Sylvester-Schur says: "if $n \ge 2k$, then there is a number in the list $n − k + 1, n − k + 2,$ ... $, n$ divisible by a prime $p > k$."

Shouldn't it also be true that if $n \ge k$, then there is a number in the list $n + 1, n + 2,$ ... $, n+k$ divisible by a prime $p > k$.

Does anyone know of a theorem that comes close to establish this or why the stronger claim is not true?

Thanks very much,

-Larry

-
Thanks for noticing! I just fixed it. :-). – Larry Freeman Nov 8 '12 at 16:32

Indeed, by a result of Denis Hanson (Canad. Math. Bull., 1973) the product (I use notation to match the papers not the question) $$\Delta(n,k)= n (n+1) \dots (n+k-1)$$ for $n \ge k$ is divisible by a prime of size greater $3k/2$ with only the exception of $3\cdot 4$ , $8 \cdot 9$ and $6\cdot 7 \dots 10$.
There are also further results in this direction. For example Laishram and Shorey (Acta Arith., 2005) proved that the largest prime divisor of $\Delta(n,k)$ is strictly greater $1.97k$ if $n \gt k+13$ and $2k$ if in addition $n> (279/262) k$