A Theorem of Sylvester and Schur

The theorem in question asserts that, if n > k, then, in the set of integers n, n+1, n-J-2,. . ., n +k-1, there is a number containing a prime divisor greater than k. If n = k+ 1, we obtain the well-known theorem of Chebyshev. The theorem was first asserted and proved by Sylvester t about forty-five years ago. Recently Schurt has rediscovered and again proved the theorem. The following proof is shorter and more elementary than the previous ones. We shall not use Chebyshev*s results, so that we shall also prove Chebyshev's theorem § .