The elementary symmetric functions of reciprocals of elements of arithmetic progressions

Let a, b and n be positive integers. Erdős and Niven [4] proved that there are only finitely many positive integers n for which one or more of the elementary symmetric functions of 1/b, 1/(a + b), . . . , 1/(an – a + b) are integers. We show that for any integer k with 1 ≦ k ≦ n, the k-th elementary symmetric function of 1/b, 1/(a + b), . . . , 1/(an – a + b) is not an integer except that either b = n = k = 1 and a ≧ 1, or a = b = 1, n = 3 and k = 2. This refines the Erdős–Niven theorem and answers an open problem raised by Chen and Tang [1].