Abstract
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].
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References
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Supported partially by National Science Foundation of China Grant #11371260 and by the Ph.D. Programs Foundation of Ministry of Education of China Grant #20100181110073.
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Hong, S., Wang, C. The elementary symmetric functions of reciprocals of elements of arithmetic progressions. Acta Math. Hungar. 144, 196–211 (2014). https://doi.org/10.1007/s10474-014-0440-2
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DOI: https://doi.org/10.1007/s10474-014-0440-2