It is well known that
multinomial coefficients are integers; i.e., if the integers ai are nonnegative and
a =∑i=1mai, then ∏i=1m(ai)!∣a!. This property may hold good in special cases
even though ∑i=1mai> a. In fact, for each integer x ≧ 0, x!(x + 1)!∣(2x)!, 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
x!(x + c)!∣(2x)!. 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 Q(x) = ((g(x) + h(x))!)∕((g(x) + k)!(h(x))!), where k is
a positive integer, and g(x) and h(x) are integer coefficient polynomials
of positive degree with positive leading coefficients and proves that some
multiple of Q(x) is integral infinitely often: a result which includes all the
earlier results. In this paper, among other things, this result of Rao has
been generalised and improved upon by taking the polynomials over the
rationals and by reducing the multiplying factor of Q(x) as obtained by
Rao.
