Vol. 17, No. 2, 1966

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A problem complementary to a problem of Erdős

J. Chidambaraswamy

Vol. 17 (1966), No. 2, 227–234

Let f(x), g(x), and h(x) be rational integer coefficient polynomials of positive degree and with positive leading coefficients and satisfying

f (x) = g(x )+ h(x ).

k(x) also being such a polynomial of degree 0, let

Q (x) = (f(x))!∕((g(x)+ k(x))!(h(x))!).

Question 1: Is Q(x) integral for an infinity of integers x, at least when k(x) of degree zero, say k(x) = k(1)?

Question 2: Is Q(x) nonintegral for all sufficiently large integers x, at least when the degree of k(x) is 1? No general answer is known to both these questions. In this paper, we consider the question of existence of an infinity of integers x for which Q(x) is not an integer: in the context of question 1, we obtain certain conditions on the coefficients of g(x) and h(x) and k to ensure the existence of an infinity of integers x for which Q(x) is not an integer, and in the context of question 2, we prove Q(x) is nonintegral infinitely often.

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