On prime divisors of the binomial coefficient

Earl F. Ecklund Jr.

Vol. 29 (1969), No. 2, 267–270

DOI: 10.2140/pjm.1969.29.267

Abstract

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 has a prime divisor p > k.

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

If n ≧ 2k, then has a prime divisor p ≦ n∕2

The problem is solved by the following

Theorem. If n ≧ 2k, then has a prime divisor p ≦ max{,}, with the exception .

Mathematical Subject Classification

Primary: 10.08

Milestones

Received: 8 July 1968

Published: 1 May 1969