Vol. 29, No. 2, 1969

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On prime divisors of the binomial coefficient

Earl F. Ecklund Jr.

Vol. 29 (1969), No. 2, 267–270

A classical theorem discovered independently by J. Sylvester and I. Schur states that in a set of k consecutive integers, each of which is greater than k, there is a number having a prime divisor greater than k. In giving an elementary proof, P. Erdös expressed the theorem in the following form:

If n 2k, then (n)
k has a prime divisor p > k.

Recently, P. Erdös suggested a problem of a complementary nature:

If n 2k, then (n)
k has a prime divisor p n∕2

The problem is solved by the following

Theorem. If n 2k, then (n)
k has a prime divisor p max{n
2}, with the exception (7)

Mathematical Subject Classification
Primary: 10.08
Received: 8 July 1968
Published: 1 May 1969
Earl F. Ecklund Jr.