Vol. 17, No. 2, 1966

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Divisibility properties of certain factorials

J. Chidambaraswamy

Vol. 17 (1966), No. 2, 215–226

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.

Mathematical Subject Classification
Primary: 10.05
Received: 6 January 1965
Published: 1 May 1966
J. Chidambaraswamy