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



Are there any theorems about a prime $p > k$ in a sequence stronger than Sylvester-Schur?
Yes, this is true. 


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 $n> (279/262) k$


