Divisibility properties of certain factorials

Abstract

CHIDAMBARASWAMY It is well known that multinomial coefficients are integers; i.e., if the integers a are nonnegative and a -^T=i ca, then S=i(;)H!. This property may hold good in special cases even though ^1 * > * n ^act ^or eacn integer x ^ 0, xl (x + 1)! I (2x)l, and it has been asked by Erdos, as a research problem in the 1947 May issue of the Monthly, whether, for a given c ^ 1, there exists an infinity of integers x such that xl(x + c)\ I (2x)l. This problem has been gradually generalized and improved upon by Mordell, Wright, McAndrew, the author, and N. V. Rao. In particular, Rao considers the quotient